Bring a large pot of salted water to a boil and cook the pasta according to the package directions. Drain, rinse with hot water, drain again, and set aside.

Pound the chicken breasts thin between two sheets of waxed paper. Slice the chicken into ½-inch strips. Combine the flour with ½ teaspoon salt and ¼ teaspoon of pepper. Toss the chicken strips in the flour mixture, coating well.

In a large skillet, heat the oil over medium-high heat. Add the butter and let it melt. Working in batches, add enough chicken to the pan so that they are in a single layer and the pan is not over-crowded. Cook until golden brown, about 2 minutes per side. Remove from the skillet. Add the mushrooms, ½ teaspoon salt, and ½ teaspoon pepper. Cook, stirring occasionally, until they release their liquid and start to brown. Add the wine and cook, stirring occasionally, until it reduces by half and starts to thicken, about 5 minutes. Add the reserved pasta and the chicken, cream, lemon juice, parsley, capers, and ½ cup Parmesan cheese. Cook, stirring, until everything is heated through and the sauce coats everything.

Serve immediately with more Parmesan.