Stir ketchup, mayonnaise, pickle relish, vinegar, white sugar, and garlic together in a small bowl. Season to taste with salt and black pepper. Cover and set aside.

Divide ground beef into 4 even pieces. Roll each piece into a ball; refrigerate until needed. Spread cut sides of each hamburger bun with butter.

Heat a large cast iron or nonstick skillet over medium-high heat. (You can add a little vegetable oil if worried about sticking.)

Add two hamburger balls to the skillet. Smash meat flat using the bottom of a small sauce pot or a spatula. Hold in place until seared, about 1 minute. Season with salt and pepper. Cook until a good crust forms, then flip.

Once the flip side looks close to being done, cover with cheese slice. Move the patties aside and place buns, butter side down, in skillet until toasted. Continue to cook burgers until cheese is melted, and meat is no longer pink in the center, about 5 minutes. An instant-read thermometer inserted into the center should read at least 160°F (70°C).