Using a teaspoon, scoop out the tomato seeds. Discard the seeds.
Chop the tomatoes into 1-inch (2.5 centimeter) chunks and transfer to a large bowl.
Add the onion, scallion, parsley, basil, salt, and pepper, and knead with one hand for about 1 minute, until the mixture gains a pulpy texture.
In a separate bowl, stir together the flour and baking powder.
Add the flour mixture to the tomato mixture and mix. Let rest for 10 minutes.
Do a taste test. In a small pan with a little oil, fry a tiny portion of the tomato mixture. Taste and adjust the seasoning, if needed.
Moisten your hands with a little water to keep the mixture from sticking to them.
Use your hands to scoop out about a heaping tablespoon of the tomato mixture, shape it into a 2-inch (5 centimeter) patty, and pack the mixture down so it won't fall apart. Place the patty on a plate or a tray. Repeat until you have used up all the tomato mixture.
Cover the bottom of a medium nonstick pan with olive oil at least 1/4 inch (6 millimeters) deep, and bring to a frying temperature over medium heat. You can check the temperature of the oil by dipping the handle of a wooden spoon in it. When the oil is ready, it will gently sizzle and bubble up around the handle.
Line a plate with a paper towel.
Working in batches, fry 3 or 4 patties for 2 to 3 minutes on each side, until golden, using two forks to turn them.
Take the patties out and let them drain on the paper towel-lined plate.
Repeat with the rest of the tomato patties, a third at a time.
Serve warm or at room temperature with tzatziki, if desired.
Leftover tomatokeftedes Santorinis can be stored in an airtight container in the refrigerator for up to 4 days. To reheat, warm in the oven at 325˚F (160˚C) for 6 to 8 minutes.