Bring a large pot of lightly salted water to a boil. Cook farfalle at a boil, stirring occasionally, until cooked through yet firm to the bite, about 12 minutes. Drain and transfer farfalle back to the pot.

Heat oil in a large skillet over medium heat; cook chicken in hot oil until no longer pink in the center and juices run clear, 5 to 8 minutes per side. Transfer chicken to a cutting board and slice into thin strips.

Melt 2 tablespoons butter in the same skillet; cook and stir onions in melted butter until translucent, 5 to 10 minutes. Stir mushrooms and chicken strips into onions; cook and stir until heated through, about 5 minutes. Remove the skillet from heat.

Melt 1/4 cup butter in another large skillet; stir flour into melted butter until dissolved, about 1 minute. Slowly whisk in milk until smooth; add garlic, salt, and pepper. Bring to a boil for 1 minute; reduce heat.

Stir chicken mixture and farfalle into garlic sauce; cook and stir over medium-low heat until heated through, 2 to 3 minutes.