Heat a gas or charcoal grill to high heat. Brush the grill grates with oil.
For the portobello mushrooms: Brush the inside of the mushroom caps with oil and season with salt and pepper. Place the mushrooms on the grill, cap-side down, and grill, covered, until golden brown and slightly charred, 4 to 5 minutes. Flip the mushrooms and cook until completely soft, another 4 to 5 minutes.
For the burgers: 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.
Brush the patties with oil and grill, covered, until slightly charred and crusty, 4 to 5 minutes. Flip the burgers and cook until slightly charred and crusty on the other side, the burgers are completely cooked through and an instant-read thermometer registers 165 degrees F, 4 to 5 minutes.
While the burgers cook, toast the buns on the grill until lightly charred, 10 to 15 seconds. Set aside.
Invert the mushrooms caps so that the inside is facing up. Place 1 slice of cheese inside each cap. Cover the mushrooms with the grill lid, large domed lid or large metal bowl and count to 10 until the cheese is melted.
Place the portobello mushroom caps on the bottom buns. Top with the turkey burgers, ketchup, freshly ground black pepper and the top buns and serve immediately.