Adjust oven rack to lower-middle position, set large heatproof plate on rack and heat oven to 200°F.
Halve 1 lemon. Trim ends from one half and cut crosswise into slices 1/8 - 1/4 inch thick; set aside. Juice remaining half and whole lemon to obtain 1/4 cup juice. Reserve.
Remove tenderloin and slice breasts in half. Sprinkle both sides of the resulting cutlets with salt and pepper. Measure flour into a pie tin or shallow dish. Working 1 cutlet at a time, coat with flour and shake to remove excess.
Heat 2 tbsp of oil in heavy bottomed skillet over medium-high heat until shimmering. Lay half of chicken pieces in skillet. Saute with moving them until lightly browned on one side (~ 3minutes). Turn and cook until second side is browned and cooked through (~3 minutes). Remove pan from heat and transfer cutlets to plate in oven. Add remaining 2 tbsp oil to skillet and repeat with remaining cutlets.
Add shallot to empty skillet, sautee until fragrant (~ 30 seconds). Add broth, lemon slices and increase heat to high, scraping bottom with wooden spoon to loosen brown bits. Simmer until liquid reduces to 1/2 cup, around 4 minutes. Add lemon juice, capers and simmer until sauce reduces again to 1/3 cup (~ 2 minutes). Remove from heat and add butter to thicken, swirl in parsley. Spoon sauce over chicken and fettucini.