In a large bowl, combine the ground chicken, breadcrumbs, Pecorino Romano, egg, parsley, shallot, capers, salt, pepper, and 1 tablespoon of olive oil. Mix gently until just combined; do not overmix.
Using a medium cookie scoop or spoon, portion the mixture and gently roll into 1½-inch meatballs. You should have approximately 14–16 meatballs.
Pour enough olive oil into a large skillet to generously coat the bottom and heat over medium heat. Working in batches, cook the meatballs for 2–3 minutes on the first side, then flip and cook for another 2–3 minutes, until golden brown and nearly cooked through. Transfer to a plate and add more olive oil between batches as needed.
Carefully drain the excess oil from the skillet, leaving about 1 tablespoon in the pan along with the browned bits.
Add the white wine and simmer for 2–3 minutes, scraping up the browned bits, until reduced by about half.
Add the chicken broth and capers. Simmer for 3–4 minutes, until the sauce is slightly reduced.
Stir in the lemon juice and butter until the butter has melted and the sauce is smooth.
Return the meatballs and any accumulated juices to the skillet. Cover and simmer gently over low heat for 3–5 minutes, until heated through and the meatballs reach an internal temperature of 165°F. Finish with fresh parsley.