In a food processor, combine basil, garlic, extra virgin olive oil, lemon juice, Parmesan, and red pepper flakes if using. Pulse until smooth but still slightly textured, scraping down the sides once. Taste and add a pinch of salt if needed. Set aside at room temperature.

In a medium bowl, toss cherry tomatoes and red onion with balsamic vinegar and 1 teaspoon olive oil. Season lightly with salt and let sit at room temperature while you cook the salmon.

Pat salmon fillets completely dry with paper towels. Season all over with kosher salt and black pepper. Heat 1 tablespoon olive oil in a large nonstick skillet over medium-high heat until shimmering. Add salmon skin-side up and cook without moving until golden brown and crisp on the bottom, about 4 minutes.

Carefully flip the fillets and reduce heat to medium. Cook until the salmon is just opaque in the center and flakes easily with a fork, 3 to 4 minutes more depending on thickness. The internal temperature should read 125F for medium.

Transfer salmon to plates. Spoon basil sauce generously over each fillet. Pile the tomato salad alongside or on top. Finish with a pinch of flaky sea salt and an extra drizzle of basil sauce if desired. Serve immediately.