Preheat oven to 350°. Prepare linguine according to package instructions. Drain and transfer to baking dish.
In a large skillet over medium heat, melt 1 tablespoon butter. Cook garlic until fragrant, 1 minute, then add mushrooms and cook until golden and moisture is released. Add chicken and cook, stirring, until lightly golden, 5 minutes. Transfer mushrooms and chicken to baking dish.
Return skillet to medium heat and melt remaining 2 tablespoons butter. Whisk in flour until completely combined, then slowly whisk in broth and bring to a simmer. Pour in heavy cream and bring to simmer over low heat, stirring occasion- ally and allowing sauce to thicken, about 10 minutes.
Pour sauce over mixture and season with salt and pepper. Toss to combine. Sprinkle with Parmesan and bake until sauce is bubbling and cheese is golden, 25 minutes.