Bring a large pot of salted water to a boil for the linguini.
Slice each chicken breast in half horizontally (if very thick), then pound to about 1/4-inch thickness between sheets of plastic wrap. Season both sides with salt and pepper.
Place flour in a shallow dish. Dredge the chicken in flour, shaking off excess.
Heat olive oil and 2 tablespoons butter in a large skillet over medium-high heat. Add chicken and cook 3–4 minutes per side, until golden and the thickest part reaches 165°F/74°C. Transfer to a plate.
Reduce heat to medium. Add garlic and cook about 30 seconds, just until fragrant.
Add chicken broth, lemon juice, and lemon zest. Scrape up browned bits, then simmer 2–3 minutes to reduce slightly. Stir in capers.
Whisk in the remaining 3 tablespoons butter until the sauce looks glossy. Return chicken to the skillet and spoon sauce over top. Simmer 1–2 minutes to warm through and meld flavors.
Meanwhile, cook linguini until al dente, then drain. Serve chicken over linguini, spoon sauce on top, and finish with chopped parsley.