Stir the mayonnaise, lemon juice, thyme and garlic together and chill the aioli.
Melt 1 tablespoon butter in a skillet over medium-high heat and cook the mushrooms for 8 to 10 minutes until browned and dry-looking.
Shape the beef into 4 thick patties with center dimples and season both sides with the salt and pepper.
Sear the patties in a hot cast-iron skillet for 4 minutes, flip them and lower the heat to medium.
Cook for 5 to 7 minutes more without pressing until the centers reach 160°F.
Top with the mushrooms and Swiss, cover and cook for 1 minute until the cheese melts around the mushrooms.
Melt the remaining butter on the cleared skillet and toast the buns cut-side down for 1 minute.
Spread thyme aioli over both bun halves and enclose one mushroom-topped patty in each.