Add ½ tablespoon butter and ½ tablespoon olive oil to a 10-inch ovenproof skillet over medium-high heat. Combine the crushed bouillon cubes and garlic powder in a small bowl. Season both sides of the chicken breasts, reserving about one-third of the seasoning mixture for later.

If your chicken breasts are large, butterfly or pound them slightly before searing so they cook more evenly. Place the chicken in the hot skillet and sear for 3 to 4 minutes per side until browned. The chicken does not need to be fully cooked at this point since it will finish in the oven. Remove the chicken from the skillet and set aside.

Add the remaining butter and olive oil to the same skillet. Stir in the mushrooms, onions, and reserved seasoning mixture. Cook over medium heat for 3 to 5 minutes, stirring often, until softened. Add the white wine vinegar and stir well.

If you do not have an ovenproof skillet, transfer everything to a foil-lined baking dish at this point. Remove the mushroom and onion mixture from the skillet temporarily. This gives you room to nestle the chicken back in cleanly. Return the chicken to the skillet and spoon the mushroom and onion mixture over the top. Sprinkle with shredded Monterey Jack cheese.