Heat a grill or grill pan to medium-high. Clean and lightly oil hot grill. In a food processor, combine mayonnaise, basil, garlic, and lemon juice, and process until basil is finely chopped and sauce is smooth (add 1 to 2 teaspoons water if needed). Cut green tomatoes into 1/2-inch-thick slices, brush with vegetable oil and season with salt and pepper. Grill until lightly charred and softened, 2 to 3 minutes per side. Transfer to a platter and spoon sauce over top.
Absolutely delicious!!! We have many green tomatoes in our garden that haven't turned yet and may not now that is starting to get colder and this is such a great use for them. I made this for a family birthday BBQ and everyone couldn't get enough!