Bring a large pot of lightly salted water to a boil. Add pasta. Cook, stirring occasionally, until pasta is hot, about 8 minutes for aldente. Drain.
Heat vegetable oil in a large skillet over medium heat; cook and stir chicken breast in hot oil until no longer pink in the center, about 10 minutes. Remove from heat.
Melt butter in another skillet over medium heat; cook and stir mushrooms and garlic until mushrooms begin to soften, about 3 minutes. Stir heavy cream into mushrooms and bring to a boil. Reduce heat to medium-low and simmer until slightly thickened, about 3 minutes. Add Parmesan and Mozzarella cheese, parsley, and salt. Continue to cook until cheese is melted, about 1 minute.
Stir pastai and chicken into mushroom-cream sauce. Simmer until heated through, about 2 minutes.