Place 1 cup plain whole-milk Greek yogurt in a small bowl and finely grate in zest from 1 lemon. Season with kosher salt and mix well. Cut lemon in half; set aside. Chill lemony yogurt until ready to serve.

Place ½ cup all-purpose flour in a shallow bowl. Lightly beat 2 large eggs in another shallow bowl to blend. Mix 1¼ cups panko, ½ cup raw white sesame seeds, and ⅓ cup everything bagel seasoning in a third shallow bowl.

Pat 4 thin Dover sole fillets (about 1½ lb.) dry and season with salt. Working one at a time, dredge fillets in flour, shaking off excess, then dip in egg, letting excess drip back into bowl. Dredge in panko mixture, pressing to adhere. Transfer fish to a baking sheet.

Pour vegetable oil into a large skillet to come ¼" up sides and heat over medium-high. Working in batches, cook fish until golden brown and crisp, 1–2 minutes per side. Transfer to a wire rack.

Cut 1 English hothouse cucumber in half lengthwise, then thinly slice crosswise. Combine cucumber, ½ medium red onion, thinly sliced, 2 cups halved cherry tomatoes, 3 Tbsp. extra-virgin olive oil, and 2 Tbsp. chopped drained capers in a large bowl. Squeeze in juice from reserved lemon halves and season with salt; toss to combine. Add ½ cup coarsely chopped dill; gently toss again. Taste and season salad with more salt if needed.

Spread some lemony yogurt on one side of each plate. Place fish alongside and top with salad; season with freshly ground black pepper.