In a 10- to 12-inch frying pan over high heat, crumble lamb with a wooden spoon, stirring often until lightly browned, about 8 minutes. Remove meat with a slotted spoon and discard all but 1 tablespoon fat from pan.
Return meat to pan; add onion and garlic. Stir occasionally until onion begins to brown, about 4 minutes. Add tomato purée, honey, and rosemary; stir occasionally until hot, about 2 minutes. Add salt and pepper to taste.
Set roll bottoms on plates and spread with goat cheese; cover with baby spinach leaves. Spoon lamb mixture over spinach and set tops in place.