Prepare a grill or turn on the broiler; the heat should be medium-high and the rack about 4 inches from the fire. Core the tomatoes and cut each horizontally into 3 or 4 thick slices. Spread them out on a rimmed baking sheet, drizzle or brush with the olive oil, turning to coat them all over, and sprinkle with salt and pepper.
The goal is to end up with tomatoes that are slightly charred and soft but not mushy.
To Grill: Transfer them to the grates over the fi re and cook, undisturbed, until they start to brown underneath and ooze some juice, 3 to 4 minutes. Carefully turn and cook them on the other side until they’re also darkened but you can still lift them from the grill without their falling apart, anywhere from 1 to 4 minutes more depending on the thickness of the slices and the heat of the grill.
To Broil: Put the baking sheet under the broiler and cook them without turning until the tops are browned and bubbling, 5 to 8 minutes. Either way, if at any point they start to look dry and stick, drizzle the tomatoes with a little more olive oil.
To serve, transfer the tomatoes to a serving platter or individual plates, sprinkle with the cheese, and serve hot, warm, or at room temperature.
Instead of Parmesan, try topping them with crumbled blue cheese or a smear of ricotta cheese.
Before serving, sprinkle about 1/4 cup chopped fresh basil, dill, parsley, or mint leaves over everything.
For an Asian twist, substitute 2 tablespoons vegetable oil mixed with 2 tablespoons sesame oil for the olive oil and use a drizzle of soy sauce instead of the Parmesan.