Preheat the oven to 400°F (200°C). Line a large sheet pan with parchment paper or aluminum foil for easy cleanup.
In a small saucepan, melt the butter over medium heat. Continue cooking, swirling the pan occasionally, until the butter turns golden brown and develops a nutty aroma, about 3-4 minutes. Remove from heat and stir in the lemon zest, lemon juice, and minced garlic. Set aside.
Place the salmon fillets skin-side down on the prepared sheet pan. Drizzle the olive oil evenly over the salmon and season each fillet with kosher salt and black pepper.
Arrange the cherry tomatoes and sliced red onion around the salmon on the sheet pan. Sprinkle the crushed red pepper flakes over the vegetables.
Pour the lemon brown butter sauce evenly over the salmon fillets and vegetables.
Bake in the preheated oven for 12-15 minutes, or until the salmon is cooked through and flakes easily with a fork. The vegetables should be tender and slightly caramelized.
Remove the sheet pan from the oven. Sprinkle the chopped parsley and dill over the salmon and vegetables.
Serve the salmon fillets with the roasted vegetables and spoon any remaining brown butter sauce from the pan over the top.