Mix ground pork and beef in a large bowl with the mashed soft garlic cloves.
Add salt (the amount depends on how salty the blue cheese is) and black pepper to taste.
Mix very well with your hands for a few minutes; this part is important so the proteins in the meats break down and bind the whole thing together. You can also knead it with a stand mixer, something I recommend if you’re making burgers for a crowd.
Divide the meat mixture into 4 equal parts. Take each part, divide it in two, and flatten each one into a disk.
Add some blue cheese and a smaller amount of mozzarella. Don't mound it much, and don't reach the edges.
Top with the remaining flat patty and carefully cover the cheese, making sure the sides are sealed. Flatten it without letting the cheese spill.
Repeat with the remaining ground meat mixture and cheese.
Place the buns in the warm oven if you want to so they're crisp when the burgers are ready.
Heat a large skillet over medium heat.
Add the patties to the skillet, and cook them for about 5-8 minutes on each side until golden brown. You might need to check inside to see if they're fully cooked. Don't cut them completely, so you don't lose the juices. Or use a meat thermometer and check the internal temperature; it should be about 160°F for well-done burgers.
Cut the hamburger buns in half and transfer each patty to a bottom bun (spread with mayo or another condiment of choice), top with some of the mushroom-onion mixture, and close with the top bun.
Serve immediately.