Halve 1 lemon lengthwise. Trim ends from 1 half, halve lengthwise again, then cut crosswise into very thin slices; set aside. Zest and juice remaining 1½ lemons to yield 1½ teaspoons zest and 3 tablespoons juice.

Add ⅓ cup oil to 12-inch nonstick skillet. Combine 2 ounces grated Parmesan, ½ cup panko, 1 lightly beaten egg, ¾ teaspoon pepper, ½ teaspoon table salt, and lemon zest in large bowl. Add 1 pound lean ground chicken and mix with your hands until thoroughly combined. Using moistened hands, pinch off and roll mixture into 24 meatballs (about 1 tablespoon each); arrange in even layer in skillet. Place skillet over medium-high heat and cook until meatballs are well browned on one side, about 4 minutes. Gently turn meatballs and continue to cook, turning as needed, until well browned on all sides, 4 to 6 minutes; transfer to plate.

Add 1 minced shallot to fat left in skillet and cook over medium heat until softened, about 30 seconds. Stir in 2 teaspoons flour and cook for 1 minute. Slowly whisk in 1 cup chicken broth, then stir in 2 tablespoons rinsed capers, lemon slices, and lemon juice and bring to simmer. Return meatballs and any accumulated juices to skillet and simmer until meatballs register 160 degrees, 3 to 5 minutes, turning meatballs halfway through simmering.

Transfer meatballs to serving platter. Off heat, whisk 3 tablespoons unsalted butter, cut into ½-inch pieces, into sauce in skillet, 1 piece at a time, until incorporated. Season with salt and pepper to taste and adjust consistency of sauce with extra broth as needed. Spoon sauce over meatballs and sprinkle with 1 tablespoon chopped parsley. Serve.