Yes. PID_V1 is exactly the PID library.
However, It might not be the PWM you are thinking about.
Provided INPUT, the current temperature, SET, the desired temperature, the PID algorithm will derive a value, ranged from 0-255.
To turn the output value into heating control, another parameter is introduced: Heating Window Size. For example, if the window size is 5000ms, and the output value from PID is 51, the controller will turn on the heating for 5000 * 51/255 = 1000ms and turn off the heating at the rest of 4000ms.
I don't have any experience about gas control. As far as I know, people use burners with pilot fire, and the way they apply PID is like controlling the heating element. When it's time to turn off the heating, the valve is turned off. The fire will be lit by the pilot fire when the valve is opened again.
Using the output value of PID to set control the opening of the valve is surely doable. You just need to interpret the PID output. ESP8266 supports software PWM. You might need to find out if the servo can be controlled by ESP8266.
I don't have experience and knowledge about it.
Some information for you:
https://lurchi.wordpress.com/2016/06/29/esp8266-pwm-revisited-and-reimplemented/
https://circuits4you.com/2017/12/21/esp8266-pwm-example/