Heat oven to 375°F. Lightly grease a 12-cup muffin tin with olive oil.
Stir ketchup, brown sugar, and vinegar in a small bowl to make the glaze. Set aside.
Combine breadcrumbs and milk in a large bowl. Let it soak for 2 minutes.
Whisk in beaten egg, ketchup, Worcestershire, mustard, onion, garlic, parsley, salt, and pepper.
Add the ground beef. Gently mix just until evenly combined; do not overwork.
Divide the mixture among the muffin cups, packing lightly. Spoon glaze over each muffin.
Bake 18–22 minutes, until the centers reach 160°F. Sprinkle cheese on top for the last 3 minutes, if using.
Rest 5 minutes. Loosen edges with a knife, lift out, and serve warm.