Put the tomato slices in a large colander and toss with a large pinch of salt. Put the colander in the sink and let the tomatoes sit for 20 minutes, then gently pat dry with a paper towel.
Meanwhile, put the flour, the egg, and the panko in three separate shallow bowls; season each with salt and pepper. Beat the egg.
Put the olive oil and butter in a large skillet over medium-high heat. Dip each tomato slice first in the flour, then in the egg, and finally in the panko, letting any excess coating drip off. Transfer half of the tomato slices to the skillet. Cook until golden brown on the bottom, 3 to 5 minutes, then turn and cook until lightly browned on the other side, another 3 to 5 minutes. Drain on paper towels. Repeat with the remaining tomato slices, flour, egg, and panko. Serve hot.
