Heat oven to 350°F (175°C). Grease or line a standard 12-cup muffin tin.
Grate the onion and carrot fine. Add to a large bowl with the beef, egg, panko, parsley, garlic, mustard, Worcestershire, paprika, salt, and pepper.
Mix by hand just until combined, about 20 seconds.
Divide among the 12 cups, filling to the rim. Brush tops with half the BBQ sauce.
Bake 15–20 minutes at 350°F (175°C), to an internal temperature of 160°F (71°C), brushing with the rest of the BBQ sauce in the last 5 minutes.
Rest 5 minutes in the tin before lifting out. Garnish with parsley or green onion.