Pan sear chicken in a saute pan in a bit of oil over medium high heat until it is lightly browned all over. Remove from the pan and drain. (The chicken does not need to be cooked through, as it will finish in the curry sauce.)
If using the same pan, wipe out any juice from the chicken. Add another TBSP of oil and when it is hot, add the curry paste and stir fry until fragrant, about 2 minutes.
Whisk in the Coconut milk a little at a time, starting with the cream at the top of the can. Allow to come to a boil before adding more of the milk, whisking constantly.
Reduce heat to medium (or medium low) and add the Kaffir Lime Leaves, 1 TBSP Fish Sauce and 1 TBSP Sugar and cook for 2-3 minutes, reducing heat so that it just simmering.
Taste the sauce and adjust if it needs a bit more Fish Sauce and/or Sugar.
At this point, if you are not ready to eat (or your rice is not ready), cover and turn off the heat until you are ready to eat.
Increase the heat to medium-high and add the chicken and mushrooms to the sauce. Cover the pan and allow to cook for 3-4 minutes until the mushrooms have released their liquid, stirring occasionally to ensure the meat doesn't stick to the pan.
Then the mushrooms have cooked, add the green beans and cook, covered, for 1 minute then add the peppers. Cook for 2-3 minutes and sprinkle with Cilantro leaves or Thai Basil leaves.
Serve immediately with Jasmine or Basmati Rice.