Pat the salmon fillets dry using paper towels before seasoning them with salt and black pepper. Dry salmon creates a crisp golden crust while cooking. Let the salmon sit at room temperature for about 10 minutes before placing it in the pan.
Heat olive oil in a large skillet over medium-high heat. Place the salmon skin-side down and cook for about 4 minutes per side until golden and slightly crispy. Once cooked, transfer the salmon to a plate and keep it warm.
Reduce the heat slightly and melt the butter in the same skillet. Add the minced garlic and sauté for about 30 seconds until fragrant. The garlic should smell aromatic but not turn dark brown.
Pour the white wine into the skillet and let it simmer for about 2 minutes. This helps lift all the flavorful browned bits from the bottom of the pan. The wine also adds brightness to the creamy sauce.
Add chicken broth and heavy cream to the skillet while stirring gently. Let the mixture simmer over medium-low heat until it starts thickening slightly. Stir occasionally so the cream stays smooth and silky.
Slowly stir in the freshly grated Parmesan cheese until fully melted into the sauce. The sauce becomes richer and creamier at this stage. Taste and adjust seasoning with salt and black pepper if needed.
Place the cooked salmon fillets back into the skillet carefully. Spoon the creamy garlic butter sauce over the top and simmer for another 2 minutes. This allows the salmon to absorb more flavor while staying moist.
Sprinkle chopped parsley over the salmon just before serving. The fresh herbs balance the richness of the sauce beautifully. Serve the salmon hot with your favorite side dishes for a complete meal.