Season chicken paillards on both sides with salt and pepper. Heat olive oil and 1 tablespoon butter in a large skillet over medium-high heat until butter foams. Add two paillards, and saute on one side until golden brown, about 2 minutes. Reduce heat to medium. Flip, and saute paillards until cooked through, about 2 minutes. Transfer to a plate. Raise heat to medium-high, and repeat with two remaining paillards.
Add shallot to skillet, and cook over medium heat, adding oil or butter as needed, stirring often, until golden, about 1 minute. Raise heat to medium-high. Add lemon segments and juice, chicken stock, and any plate juices, and deglaze the pan, scraping brown bits from bottom with a wooden spoon. Simmer until sauce reduces by half, about 3 minutes. Gradually stir in remaining 2 to 3 tablespoons butter until just melted. Season to taste.
Other good flavorings for chicken include mushrooms, garlic, capers, Dijon mustard, tarragon, parsley, and chives. Try deglazing the pan with white wine or vermouth.