Heat a gas or charcoal grill to high heat. Place a large heatproof skillet on the grill (alternatively, you can use a stovetop over medium heat). Place 4 cazuelas on the grill for a few minutes.
For the mushrooms: Heat the olive oil in the large heatproof skillet until it begins to shimmer. Add the mushrooms and some salt and pepper and cook, stirring just a few times, until they release their liquid and turn golden brown, about 10 minutes.
Stir in the garlic and oregano and cook for 1 minute longer. Remove from the heat and set aside.
For the burgers: Combine the ground beef and pork together in a large bowl, being careful not to overmix. Divide the meat into 4 equal portions (about 6 ounces each). Shape each portion with your hands into a round patty, about 1 1/2 inches thick. Use your thumb to make a divot in the center of each burger. Sprinkle both sides of each burger with salt and pepper.
Add enough olive oil to each cazuela to thinly coat the bottom and sides. Carefully place each patty, divot-side up, inside each cazuela. Combine the fennel, paprika and some salt and pepper in a small bowl, then sprinkle over top of each burger. Grill, covered, until charred and completely cooked through, 3 to 4 minutes per side.
While the burgers cook, toast the buns on the grill until lightly charred, 10 to 15 seconds. Set aside.
Add the cheese to the tops of the burgers. Cover the burgers with the grill lid, large domed lid or large metal bowl and count to 10 until the cheese is melted.
Top each burger with a large spoonful of the mushrooms. Place the burgers on the bottom buns. Top with the ketchup, oregano and top buns and serve immediately.