Pat the chicken breasts dry with paper towels. Season both sides generously with salt, pepper, and lemon pepper if you like an extra zing. Dredge each breast in the all-purpose flour, shaking off any excess.
Bring a pot of salted water to a boil and cook the angel hair pasta according to package directions, usually 3-4 minutes for al dente. Drain the pasta and set aside, reserving a small cup of pasta water to loosen the sauce if needed.
Heat the olive oil and 2 tablespoons of butter in a large skillet over medium-high heat. Once hot, add the chicken breasts and cook until golden brown on one side, about 4 minutes. Flip and cook the other side until cooked through, about 3-5 more minutes. Remove the chicken from the skillet and tent with foil to keep warm.
Reduce heat to medium. Add the minced garlic to the skillet and sauté until fragrant, about 30 seconds. Pour in the chicken broth, fresh lemon juice, and lemon zest. Bring the mixture to a simmer, scraping up any browned bits from the bottom of the pan. Stir in the capers and remaining butter, whisking until the butter melts and the sauce thickens slightly.
Return the chicken breasts to the skillet, spooning the sauce over each piece. Let them warm through for a minute or two. Toss the cooked angel hair pasta in the sauce or serve it alongside the chicken, garnished with fresh parsley and lemon slices.
