Slice each chicken breast horizontally through the middle to create two thin cutlets, then pound each cutlet to 1/4 inch thickness between plastic wrap.
Season the cutlets with salt and pepper, then dredge both sides in flour mixed with 3/4 teaspoon salt and 1/2 teaspoon pepper, shaking off excess.
Cook angel hair pasta in generously salted water according to package directions until al dente, then drain and set aside.
Heat olive oil in a large skillet over medium-high heat and sear the chicken in batches for 3-4 minutes per side until golden brown and cooked through. Transfer to a plate and keep warm.
Reduce heat to medium, add butter to the skillet, and cook mushrooms in a single layer for 3-4 minutes until golden brown. Transfer to the plate with the chicken.
Add olive oil to the pan and cook shallot for 2 minutes until softened, then add garlic and cook 30-45 seconds until fragrant.
Pour in white wine and scrape up browned bits from the pan, letting it simmer for 2 minutes until reduced by half.
Add chicken broth and lemon juice, bring to a simmer, and cook for 3-4 minutes until slightly reduced. Taste and adjust seasoning.
Stir in heavy cream and simmer for 2-3 minutes until thickened. Add rinsed capers and swirl in the remaining tablespoon of butter until melted.
Return the seared chicken and mushrooms to the skillet, spooning sauce over the top. Simmer for 1-2 minutes until warmed through.
Serve the chicken over angel hair pasta with plenty of sauce, topped with Parmesan cheese, parsley, and lemon slices for garnish.