Heat 1 tablespoon butter and 1 tablespoon vegetable oil in a large, wide skillet or wok over medium-high heat.
Add chicken breast, salt, and black pepper, and cook on high heat for 5-6 minutes until the chicken is lightly browned and cooked thoroughly.
Now add the remaining 1 tablespoon butter and 1 tablespoon vegetable oil, onion, garlic, mushrooms, red bell pepper, yellow bell pepper, and soy sauce to the skillet and mix well.
Cook on high heat until all the liquid has evaporated, vegetables are slightly cooked, and the chicken is almost dry (4-5 minutes).
Adjust the seasoning and serve hot. You can garnish the dish with some chopped spring onion greens (scallions) and toasted sesame seeds.