Bring a large pot of lightly salted water to a boil; cook the linguine at a boil until tender yet firm to the bite, about 11 minutes; drain.
Heat butter and 1 tablespoon olive oil in a skillet over medium heat; cook and stir onion, mushrooms, and garlic in the hot oil-butter mixture until slightly softened, 2 to 3 minutes.
Heat 1 teaspoon olive oil in a separate skillet; cook and stir chicken in the hot oil until no longer pink in the center, about 10 minutes. Transfer chicken to onion mixture.
Pour white wine into chicken-onion mixture; bring to a simmer. Add tomatoes, parsley, red pepper flakes, and linguine; mix well. Simmer until heated through, 3 to 5 minutes. Squeeze lemon halves over mixture; toss.