Something similar could be built easily using an ESP8266, but a single AA battery would not be enough. Looking at this guide[1] it looks like new sw can already be flashed on it, but other than using the uart or some gpio, i don't see attempts at using the wifi interface.
Amazon's WiFi Button is way way better than the ESP8266: it has 1MBit internal flash, 16MBit external flash, a powerful 120MHz microcontroller with tons of peripherals (2 DACs, 3 ADCs, etc), an internal MEMS microphone (!)
The microcontroller on Amazon's button also has a real 1500-page documentation in English, instead of 10-page badly translated Chinese datasheet.
[1] https://learn.adafruit.com/dash-hacking-bare-metal-stm32-pro...