Pat the salmon fillets completely dry using paper towels. Season all sides generously with sea salt and coarsely cracked black pepper. Set aside.

In a deep skillet or an oven-safe ceramic baking dish, melt the unsalted butter and extra virgin olive oil together over medium-low heat. Add the sliced garlic, diced shallots, thyme sprigs, and rosemary sprigs. Cook gently for 2–3 minutes until the aromatics are soft and highly fragrant, ensuring the garlic does not brown.

Toss the whole cherry tomatoes into the aromatic butter mixture. Pour in the broth and fresh lemon juice. Simmer gently on the stovetop for 4–5 minutes until the tomato skins begin to soften and expand.

Move the tomatoes slightly to the sides of the dish. Nestle the seasoned salmon fillets directly into the liquid. Spoon a generous amount of the warm garlic butter sauce over the tops of the fillets.

Transfer the entire dish into an oven preheated to 350°F (175°C). Bake uncovered for 12–14 minutes. Halfway through the baking time, quickly baste the tops of the fish with the pan juices.

Remove from the oven when the salmon is opaque on the outside and flakes gently with a fork. Arrange the fresh lemon rounds around the fish, garnish with a handful of fresh thyme leaves, and let rest for 2 minutes before serving.