somehow your description is a bit misleading:
"The oil flow is controlled by an electric regulator and is either high fire, or low fire. The default is low fire, but when the thermostat detects that the temperature has dropped it sends a signal to the regulator to turn to high fire, when the temperature is reached it switches back to low fire and so on maintaining the temperature range."
you are linking to a thermostat, with a connector of 2 pins.
so the AGA must have a low fire at all times, and when the thermostat are activated (when temp is lower than set temp on the knob) it start high fire.
so basically you just need a relay that can withstand the current of the oilburner. check on the current thermostat it must state the current it can withstand, then a DS18b20 and an Ardino connected to your favorite HomeAutomation controller (domoticz, homeassistand, HomeSeer)
And then you can swith on/off from homeautomation from your phone
Just remember to create the code so you measure temperature and relay must switch OFF at a set temperature so you wont overheat the AGA and potentially burn your house down