Slice the tomatoes into 1/2-inch rounds. Choose firm, unripe green tomatoes so the slices hold their shape.
Set out three shallow bowls. In the first, whisk 1/2 cup of the flour with 1/4 teaspoon salt and 1/8 teaspoon pepper.
In the second, whisk the egg and buttermilk.
In the third, stir together the cornmeal, the remaining 1/4 cup flour, and the remaining salt and pepper.
Working one slice at a time, dredge in the flour, dip in the egg mixture and let the excess drip off, then coat in the cornmeal mixture.
Set on a wire rack over a baking sheet and refrigerate 15 to 30 minutes to help the coating stick.
Pour oil into a large cast iron skillet or Dutch oven to a depth of about 1 inch. Heat over medium to 350°F - 375°F.
Working in batches and without crowding the pan, fry the slices until golden and crisp on the first side, about 3 minutes, then flip gently and cook until golden on the second side.
Transfer to a paper-towel-lined plate, season lightly with salt, and serve hot. Add ranch or remoulade for dipping if you like.