Run your hands under very cold water for a minute — this will keep the meat from getting gummy — then gently combine the two meats. Divide the meat into four equal portions (six ounces each), then gently form into patties that are 1¼ inches thick and three inches in diameter. Season each patty all over — top, bottom, and the circumference — with ½ teaspoon salt and ½ teaspoon black pepper. Touch the patties as tenderly and as little as possible — the more you handle and compact the meat, the tougher it becomes.
Heat a cast-iron skillet on low heat for two minutes, add the olive oil (this prevents excessive splattering), then increase the heat to medium, medium-high, and place patties in pan. Cover with a splatter screen, if you own one, to minimize the mess on your stove top. Cook for seven minutes on one side, flip, and cook for five more minutes. Do not turn, touch, press down on, or otherwise molest the burgers while they are cooking. Place cheese on top of the second side immediately upon flipping or place in a hot oven or under the broiler until the cheese is just melted but not liquefied once the burger is cooked to your liking. (the time above is for a medium-rare to medium burger)
Turn on the broiling mechanism in your oven. Slice four the Brioche buns, drizzle with olive oil, and place on the top rack in your oven. Broil until golden brown - about 2-3 minutes (but be sure to check frequently as they will burn quickly).
Generously smear both the tops and the bottoms (or just one) with the room-temperature garlic herb butter. (The remaining butter can be refrigerated for up to a week or frozen for up to six weeks — it is delicious on everything from toast to steak.) Place the burger on the bottom bun, dress with relish, top with the bun, and enjoy!