Core and cut tomatoes into ¼-inch thick slices, lay on a baking sheet and sprinkle both sides generously (like your seasoning a steak) with salt and sugar. Let sit for 30 minutes.
Add beaten egg to a wide shallow bowl. Combine panko and Parmesan in another wide shallow bowl. Set both near the stove. Press the tomatoes between two paper towels to dry them. Season the slices again with a little salt. Dip each tomato slice in egg, then panko-Parmesan mixture, press panko mixture deeply into each tomato slice. Set the tomatoes on a clean baking sheet.
Heat sauté pan over medium-high heat, and add the olive oil and butter. When the foam subsides, add the tomatoes, enough to cover the base of the pan in a single layer. Let sit, check when you see them begin to brown (you want a nut brown crust.) Turn and brown the other side. Keep working in batches, adding more oil and butter to the pan as needed, and transfer the tomatoes to a warm serving platter. Grind (coarsely!) fresh pepper on top, sprinkle with extra cheese, and serve!
