Heat the oven to 220 C / 425 F and line a 33 x 46 cm / 13 x 18 inch rimmed sheet pan with parchment. Grate the cucumber into a 20 cm / 8-inch sieve set over a bowl, toss with 1/4 tsp salt, and let it drain for 10 minutes while you mix the meatballs.
Put the panko and egg in a 3-quart / 2.8 L mixing bowl and let them sit for 2 minutes. Add the beef, feta, grated onion, garlic, parsley, lemon zest, oregano, cumin, 3/4 tsp salt and the pepper. Mix with your hands for about 30 seconds, only until no streaks of egg remain.
Scoop 40 g portions, about the size of a golf ball, and roll them smooth. Set them 2.5 cm / 1 inch apart on the pan to make about 20 meatballs, then brush the tops with 1 tbsp olive oil.
Bake for 18 to 20 minutes, until the tops are browned and a meatball from the center of the pan reads 160 F / 71 C. Wrap the pita in foil and warm it on the lower rack for the last 5 minutes.
While the meatballs bake, squeeze the drained cucumber in a clean kitchen towel until nothing drips. Stir it into the whole-milk plain Greek yogurt with the garlic, 1 tbsp olive oil, lemon juice, dill and the remaining 1/4 tsp salt, then chill until serving.
Rest the meatballs on the pan for 5 minutes. Toss the tomatoes with the red onion, 1 tbsp olive oil, vinegar and a pinch of salt. Plate 5 meatballs per person with a warm pita, about 1/2 cup tzatziki and the tomato salad.