This recipe yields more pesto than you need for the tomatoes, but why make just a little? You will have enough pesto left over to sauce a three-quarters of a pound of pasta the next day, a head start on dinner for four. Serve the tomatoes with grilled fish or steak.
Prepare a moderate charcoal fire or preheat a gas grill to medium (375 degrees).
To make the pesto, put the basil, garlic, and pine nuts in a food processor and pulse until well chopped. With the motor running, add the ½ cup of olive oil gradually through the feed tube, stopping once or twice to scrape down the sides of the work bowl. Purée until the pesto is almost, but not completely smooth. Transfer to a bowl and stir in the cheese. Season to taste with salt.
Core the tomatoes. Slice off the rounded top and bottom of each tomato, then cut the remainder into ¾-inch-thick slices. Brush with the remaining 1 tablespoon of olive oil on both sides and sprinkle on both sides with salt.
Grill the tomatoes directly over the coals or gas flame, turning once, just until hot throughout, 2 to 3 minutes per side. Do not overcook or they will become mushy. Transfer the tomato slices to a serving platter and slather them with the pesto, using 1 heaping teaspoon per slice (reserve the remaining pesto for another use), and serve immediately.