Using a serrated knife, slice the green tomatoes to a 1/4-inch thickness. Sprinkle the tomatoes with a bit of salt and allow them to sit on a few paper towels to drain.

Now we're going to make our assembly line. In one bowl, add the flour. In another, whisk together the cornmeal, bread crumbs, salt, cayenne and black pepper. Lastly, whisk the egg in another bowl (you may need another egg--add it later if you need to).

Heat a few tablespoons of oil in a pan that is set over moderately medium heat. Dip the slices of tomatoes in the flour, and then the beaten egg and then, for its last trip, to the bowl with the bread crumbs. Make sure the tomatoes are completely coated, sides of the tomatoes included! Don't be afraid to use your other hand to sprinkle the bare spots, if needed.

Transfer the tomatoes to the hot oil and fry on each side for about 1 minute, until crispy and golden brown. Place them on a bed of paper towels to drain. Add any more oil to the pan if needed and continue with frying the remaining slices up.

To make the grilled cheese, smear a few tablespoons of pimento cheese onto each piece of bread, top with a handful of grated cheese and a few fried green tomatoes. Top it with another piece of bread. Repeat with the remaining sandwich. In a saucepan (you can even use the one you used to fry the fried green tomatoes), heat a tablespoon of butter over medium heat. When the pan is hot, add the sandwich, flipping at when the underside is crispy and golden brown. Turn the sandwich onto its opposite side and cover the pan to help melt the cheese. Remove and repeat with the other sandwich. Slice in half and serve.