Split the croissants horizontally into tops and bottoms, optionally warming store-bought mini croissants slightly in the oven first. The pastry should feel firm enough to support the filling without breaking apart.

Stir the mayonnaise, yellow mustard, sweet relish, white vinegar, garlic powder, onion powder, paprika, salt, and pepper together in a bowl, then set it aside. The mixture should look evenly blended, with relish distributed throughout the creamy sauce.

Blend the melted butter with the garlic, shredded Parmesan, salt, and black pepper in another bowl, then set it aside. Stir until the garlic and cheese are dispersed so each brushful carries seasoning.

Loosely shape the beef into small oval portions sized for the croissants and season them with salt and pepper. The portions should hold together without feeling tightly packed.

Heat a large cast-iron skillet or griddle over high heat, then add a batch of beef portions with space around each and immediately flatten them with a spatula or burger press. A sharp sizzle and broad contact with the pan help develop a thin patty with irregular crisp edges.

Leave the patties undisturbed for 2-3 minutes without flipping them yet. Look for browned, crisp borders as the exposed edges turn gray, but don't use color alone to judge doneness.

After the initial 2-3 minutes, place cheese on the patties if using, cover the pan for 1 minute, and check that the beef is cooked through before assembly. The trapped heat should soften the cheese into a glossy layer while the patties finish cooking.

Heat another large pan over medium heat, brush garlic butter onto the outside bottom of each croissant, and place the bottom halves buttered-side down. The butter should sizzle gently rather than darken immediately.

Layer sauce, a cooked patty, more sauce, and the top croissant half over each bottom, then brush the outside top with garlic butter and press gently with the flat back of a spatula. Use enough pressure for steady pan contact without crushing the pastry layers.

Turn each assembled burger, gently press the other side, and remove it when both outer surfaces are toasted. Look for golden pastry and a buttery, toasted aroma without dark garlic flecks.

Transfer the burgers to plates and serve immediately. The croissants should still feel crisp outside, with warm beef and melted cheese inside.