In a medium bowl combine the ground beef, dry breadcrumbs, milk, Worcestershire sauce, kosher salt, and black pepper. If using grated carrot or chopped onion, add them now. Gently mix with your hands or a spoon until just combined — do not overwork the meat or patties will become dense.

Portion

Divide the mixture into 4 equal portions (about 4 ounces each). Shape each portion into a slightly flattened round patty about the diameter of the mini buns and roughly 1/2 inch thick. Press a small shallow indentation in the center of each patty with your thumb to help them keep shape while cooking.

Cook

Heat a large nonstick or cast iron skillet over medium-high heat. Add 1 teaspoon vegetable oil and swirl to coat. When oil is shimmering, add the patties (work in a single layer, don’t overcrowd). Cook for 3 minutes on the first side without moving to get a good sear. Flip and cook 2–3 minutes on the second side for a medium doneness. In the final 30–45 seconds, place a half-slice of American cheese on each patty and cover the skillet with a lid to melt the cheese. Internal temperature should reach 160°F (71°C) for ground beef.

Toppings & Assembly

Toast buns

While patties are cooking, split the mini buns and spread a thin layer of softened butter on cut sides. Place cut sides down in a separate dry skillet over medium heat for 30–60 seconds until lightly golden, or toast in a 350°F (175°C) oven for 4–6 minutes. Watch closely so they don’t burn.

Prepare sauce

In a small bowl combine the mayonnaise, ketchup, and yellow mustard. Stir until smooth — this creates a kid-friendly burger sauce. Taste and adjust quantities to preference.

Assemble burgers

Spread about 1 teaspoon of the sauce on the bottom of each bun. Place a lettuce leaf on top of the sauce, then a slice of tomato (if using). Add a cheesy beef patty, then two pickle slices. Spoon a little more sauce on the top bun if desired and close the burger. Secure with a toothpick for easier handling if desired.

Serve

Arrange Kidd Burgers on a platter and serve warm with optional sides such as oven fries, apple slices, or carrot sticks. Leftover patties keep refrigerated for up to 3 days; reheat gently in a skillet or microwave.