Season both sides of the salmon fillets generously with salt and pepper.

Heat a large skillet or cast iron pan over medium-high heat. Add the olive oil and pan-fry the salmon until deep golden brown on both sides, about 4 minutes per side. Drain most of the excess oil, leaving about 1 teaspoon in the pan.

Add the butter to the pan and let it melt. Tilt the pan toward you and baste the salmon repeatedly with the hot butter for about 30 seconds. Remove the salmon from the pan and set aside on a plate.

Saute the minced garlic over medium heat for about 30 seconds until fragrant and just golden at the edges. Do not allow it to brown.

Optional: Add the white wine and simmer for about 2 minutes until mostly evaporated.

Reduce heat to medium-low. Pour in the chicken broth and heavy cream. Add the grated Parmesan and stir steadily until the cheese fully melts and the sauce turns smooth and glossy. If the sauce looks like it is separating, lower the heat immediately and stir continuously. Season with parsley, salt, and pepper to taste.

Simmer the sauce for 2 to 3 minutes until it thickens enough to coat the back of a spoon. Return the salmon to the pan and spoon the sauce generously over the fillets. Simmer for 2 to 3 more minutes until the salmon is warmed through. Do not overcook.

Plate the salmon, spoon extra sauce on top, and garnish with fresh chopped parsley. Serve immediately.