Heat a skillet over medium-high heat. Add the olive oil and butter. When melted, add the onion and cook until translucent, about 5 minutes. Add the mushrooms, sea salt, and pepper, to taste. Cook until the mushrooms are tender, about 5 minutes. Remove the pan from the heat and allow to cool. In a large bowl, combine the cooled onion mixture and ground beef with sea salt and pepper to taste. To adjust seasonings, form a small test patty, cook, and taste. Adjust if necessary. Form the meat mixture into 6 patties.
Place the patties on the heated grill and cook, turning once, until desired doneness, about 2 to 3 minutes per side for medium-rare.