I can't think of any organized way to walk through what I did to get to the point I am right now, so I'm just going to put the YAML here and let the comments speak for themselves. I wish the syntax highlighting was nicer, but there it is.

Based on the YAML I found for the M5StickC Plus on the ESPHome devices page:

substitutions:
  devicename: test-dongle
  upper_devicename: DONGLE TESTING

esphome:
  name: $devicename
  platform: ESP32
  board: m5stick-c    #there are only a few options, and this is the closest
  platformio_options:
    upload_speed: 115200

wifi:
  ssid: !secret wifi_ssid
  password: !secret wifi_password
  
  ap:
    ssid: $devicename Fallback Hotspot
    password: !secret wifi_password

captive_portal:

# Enable logging
logger:

# Enable Home Assistant API
api:
  encryption:
    key: "<redacted>"

ota:
  password: "<redacted>"

# The older version of this dongle has an AWP192 power management chip, so
# the YAML I started with had support for reading the battery voltage 
# from the PM chip. The AWP192 was also needed to turrn the LCD backlight
# on for some reason. The PLUS2 stick doesn't have a power management chip,
# so we just read the battery voltage from the ADC on GPIO38 (got that
# little tidbit from diving into the schematic.)
sensor:
  - platform: adc
    pin: GPIO38                # ADC1_CH2
    attenuation: 11db          # because reasons?  
    update_interval: 60s
    name: "Battery Voltage"
    filters:
      - multiply: 2.0          # Schematic shows a voltage divider with
                               # two 100k resistors feeding GPIO38, so
                               # we multiply by two. Seems to be giving
                               # accurate readings -- or at least reasonable  

# Built-in 6-axis intertial measurement unit (IMU) that also includes a temperature sensor
# Pretty much just pulled this from the ESPHome docs, seems to work well
  - platform: mpu6886
    i2c_id: bus_a
    address: 0x68
    update_interval: 10s
    accel_x:
      name: "MPU6886 Accel X"
    accel_y:
      name: "MPU6886 Accel Y"
    accel_z:
      name: "MPU6886 Accel z"
    gyro_x:
      name: "MPU6886 Gyro X"
    gyro_y:
      name: "MPU6886 Gyro Y"
    gyro_z:
      name: "MPU6886 Gyro z"
    temperature:
      name: "MPU6886 Temperature"

# Button setup. The dongle has three buttons, this code does useful(ish)
# things with them:
#        Button A (big one on front): Turns off LCD backlight while pressed
#        Button B (small one on side): Sounds 1000Hz tone while pressed
#        Button C (other small one): Supposed to turn off the internal LED, 
#                                    but doesn't work
binary_sensor:
  - platform: gpio
    pin:
      number: GPIO37
      inverted: true
    name: ${upper_devicename} Button A
    on_press:
      then:
        - light.turn_on: display_bl
    on_release:
      then:
        - light.turn_off: display_bl

  - platform: gpio
    pin:
      number: GPIO39
      inverted: true
    name: ${upper_devicename} Button B
    on_press:
      then:
        #- light.turn_on: led1
        - output.turn_on: buzzer
        - output.ledc.set_frequency:
            id: buzzer
            frequency: "1000Hz"
        - output.set_level:
            id: buzzer
            level: "50%"     

    on_release:
      then:
        #- light.turn_off: led1
        - output.turn_off: buzzer
  
  - platform: gpio
    pin:
      number: GPIO35
      inverted: true
    name: ${upper_devicename} Button C
    on_press:
      then:
        - light.turn_on: led1
    on_release:
      then:
        - light.turn_off: led1

# Built-in LED is on GPIO19. The IR LED is also on the same GPIO. One or 
# both of them get VERY hot while LEDs are on.
light:
  - platform: monochromatic
    output:  builtin_led
    name: ${upper_devicename} Led
    id: led1

  - platform: monochromatic
    output:  backlight
    name: ${upper_devicename} Backlight
    id: display_bl    
output:
  - platform: ledc
    pin: 19
    inverted: true
    id: builtin_led

  - platform: ledc        # Buzzer is connected to GPIO2, and we 
    pin: 2                # use PWM to control it, hence the ledc  
    inverted: true        # platform
    id: buzzer 

  - platform: ledc        # TFT backlight is on GPIO19, ditto PWM
    pin: 27
    inverted: true
    id: backlight    

# This block was what was killing the original compile, haven't played
# with it yet
#remote_transmitter:
#  - pin:
#      number: GPIO9
#   carrier_duty_percent: 50%
#    id: internal

# From the schematic it looks like the TFT is using SPI 
spi:
  clk_pin: GPIO13
  mosi_pin: GPIO15

# MMU uses I2C
i2c:
   - id: bus_a
     sda: GPIO21
     scl: GPIO22
     scan: True

# Get something on the display
font:
  - file: "gfonts://Roboto"
    id: roboto
    size: 18

color:
  - id: my_white
    red: 100%
    green: 100%
    blue: 100%

# 1.14 inch, 135*240 Colorful TFT LCD, ST7789v2
display:
  - platform:  st7789v
    model: TTGO TDisplay 135x240
    cs_pin: GPIO5                     # Had to change pin assignments 
    dc_pin: GPIO14
    reset_pin: GPIO12
    rotation: 270
    lambda: |-
     it.print(80, 0, id(roboto), id(my_white), TextAlign::TOP_CENTER, "M5Stick Test");

# Haven't worked on audio yet

#i2s_audio:
#  id: bus_i2s
#  i2s_lrclk_pin: G26
#  i2s_bclk_pin: G0

#microphone:
#  - platform: i2s_audio
#    i2s_din_pin: GPIO34
#    i2s_audio_id: bus_i2s
#    adc_type: external
#    pdm: true
#    id: mic