Preheat oven to 400°F; line a sheet pan with parchment or lightly grease it.
Gently mix beef, breadcrumbs, egg, onion, garlic, salt, pepper, and 2 tablespoons BBQ sauce in a large bowl just until combined; don't overmix.
Divide in half; shape into two oval loaves about 4 inches long and 2 inches tall. Place on one side of pan, spaced apart.
Toss potatoes with 1 tablespoon oil, paprika, and a pinch of salt in a medium bowl; arrange on other side.
Generously brush loaf tops and sides with half the remaining BBQ sauce.
Bake 20 minutes; remove pan. Toss beans with remaining 1 tablespoon oil; nestle around potatoes.
Brush loaves with remaining sauce; bake another 12-15 minutes until meat reaches 160°F internally and vegetables are tender and lightly browned.
Rest loaves 5 minutes; serve with potatoes and beans.