In a large bowl, combine ground chicken, breadcrumbs, Parmesan, egg, parsley, shallot, capers, salt, pepper, and 1 tablespoon olive oil. Mix gently until just combined — do not overmix.
With a small scooper, portion the mixture and roll into 18–20 meatballs, about 1 ½ inches each.
Heat 3–4 tablespoons olive oil in a large skillet over medium heat. Add half the meatballs in batches, searing until golden brown, about 5-7 minutes. Transfer to a plate. (They don't need to be fully cooked yet.) Add more olive oil if needed and repeat with the second batch.
After frying, carefully drain off excess oil from the skillet, leaving about 1 tablespoon along with the browned bits.
In the same skillet, melt the butter. Stir in the white wine and simmer for 2–3 minutes, scraping up browned bits, until reduced slightly.
Add chicken broth and lemon juice. Bring to a gentle simmer and let the sauce reduce for 3–4 minutes so it begins to thicken.
Stir in the capers, then return the meatballs to the skillet. Cover and simmer gently 5–7 minutes, until the sauce thickens and the meatballs reach 165°F internal temperature.
Sprinkle with fresh parsley before serving.