Pound each chicken breast between plastic wrap to an even ½-inch thickness. Season both sides with salt and pepper, then dredge lightly in flour.
Heat the olive oil and 1 tablespoon of butter in a large skillet over medium-high heat. Sear the chicken 4–5 minutes per side, until golden and cooked through to 165°F. Remove and tent loosely with foil.
Meanwhile, bring a large pot of salted water to a boil and cook the farfalle 1 minute less than the package instructions. Drain and set aside.
In the same skillet, cook the prosciutto 1–2 minutes per side until crisp. Remove and roughly chop once cool.
Add the remaining tablespoon of butter to the skillet. Sauté the mushrooms 5–6 minutes until browned. Add the shallot and garlic and cook 1 more minute.
Pour in the white wine and simmer 2 minutes, scraping the pan bottom. Add the chicken broth and sun-dried tomatoes, then stir in the heavy cream.
Simmer for about 10 minutes, stirring occasionally, until the sauce coats the back of a spoon and holds a clean line for a couple of seconds.
Remove from heat and stir in the parmesan until smooth. Slice the chicken and return it to the pan with the chopped prosciutto.
Add the drained pasta and toss over low heat for 1–2 minutes, until coated and the pasta finishes cooking.
Serve immediately, topped with extra parmesan and torn basil.