Add the garlic cloves and oil to a small saucepot and place over medium-high heat. Cook until the garlic is golden brown and soft, 10 to 15 minutes. Remove the garlic from the oil, reserving the oil.
Place the garlic cloves into a mini food processor with the mayo and 1 teaspoon salt. Puree until smooth. Transfer to a small bowl.
Use a wide-mouth Mason jar lid to form the ground beef into six patties, about 5.5 ounces each.
Heat a large cast-iron skillet over high heat.
Brush the patties on one side with garlic oil, and season with salt and 1/4 teaspoon black pepper.
Place the patties, brushed-side down, in the hot skillet and cook until well browned, 5 minutes. Brush the other side of the patties with garlic oil and season with more salt and pepper. Flip the patties, and cook until well browned and firm to the touch, another 3 to 5 minutes; place a slice of Cheddar on each burger during the last minute to melt.
Spread the brioche buns with softened butter and brown them in a large saute pan.
Spread the toasted buns with some garlic mayo, add a burger patty and top with a tomato slice, lettuce and pickles. Serve immediately.