Pat the salmon fillets dry with paper towels on both sides, then season them evenly with about half of the salt and half of the black pepper.
Heat the olive oil in a large, deep skillet or wide pan over medium heat until it shimmers.
Place the seasoned salmon fillets in the hot pan, skin-side down if using skin-on, and sear for 3 to 4 minutes until golden around the edges.
Carefully flip the salmon and cook for another 2 to 3 minutes, just until lightly browned on the second side, then transfer the fillets to a plate and set aside; they will finish cooking in the sauce.
Reduce the heat to medium-low, then add the unsalted butter to the same pan and let it melt, scraping up any browned bits from the bottom with a wooden spoon.
Stir in the minced garlic and grated ginger, cooking for about 1 minute until fragrant, stirring frequently so the garlic does not burn.
Add the red curry paste, ground turmeric, and ground cumin to the pan, stirring them into the garlic butter for 1 to 2 minutes to toast the spices and deepen their flavor.
Pour in the coconut milk and vegetable or chicken broth, whisking or stirring well until the curry paste and spices are fully blended into a smooth sauce.
Season the sauce with the remaining salt and black pepper, then bring it to a gentle simmer over medium heat.
Stir in the fresh lime juice and the halved cherry tomatoes, letting the sauce bubble softly for 3 to 4 minutes so it thickens slightly and the tomatoes begin to soften.
Nestle the seared salmon fillets back into the pan, spooning some of the coconut curry garlic butter sauce over the top of each piece.
Simmer the salmon gently in the sauce for 5 to 7 minutes, or until the fillets are just cooked through and flake easily with a fork, adjusting the heat as needed to keep the sauce from boiling hard.
If using crushed red pepper flakes, sprinkle them over the sauce during the last minute of cooking and taste the sauce, adjusting salt or lime juice as needed.
Remove the pan from the heat and scatter the chopped fresh cilantro over the salmon and sauce just before serving.