Slice the tomatoes crosswise into 1/2-inch slices.
In a shallow bowl, combine the cornmeal, salt, and cayenne pepper. In another shallow bowl, beat the eggs. Line a surface or large plate with paper towels. Dip each tomato slice into the eggs, lift to allow excess egg to drip off, then place them in the cornmeal mixture and turn so that they are coated with the crumbs. Place on a wire rack as they are finished.
Fill a large skillet with oil to a depth of about 1/4 inch. Heat over medium-high heat until barely shimmering. Add the tomatoes in a single layer without crowding them (you will probably need to do this in batches). Fry them for about 2 minutes until they are golden on the bottom, then flip them and fry for another 2 minutes, until the other side is golden. Transfer the tomatoes to the paper towel-lined plate or surface. Repeat until all of the tomatoes are cooked.
Serve the tomatoes as hot as you can. If desired, serve with the sauce of your choice on the side for drizzling over.
Notes
The key to crispy fried green tomatoes is not to crowd the pan. Place only a few tomatoes in the oil at a time, in a single layer with some space between them. This means every bit of the tomato's surface will have contact with the oil, getting crispy on all sides.