Melt butter over medium heat. Add the garlic and sauté 1-2 minutes or until fragrant.
Add chicken broth, lemon juice, and salt to taste. Simmer this mixture until reduced by 1/3 to 1/2. The sauce should be thick and bubbly.
Remove from heat and set aside.
Salmon
Remove the salmon from the fridge 10-20 minutes before cooking. Pat dry.
Season both sides of the fish with salt and pepper, if desired.
Heat olive oil in a large saucepan over medium-high heat until shimmering.
Place salmon fillets skin side up and cook about 2-3 minutes or until light brown and crispy. Using a spatula, carefully flip them over.
Cook 3-4 minutes on second side, or until skin is crisp and flesh begins to feel firm to the touch. Remove at this point for a salmon cooked around medium.
Continue to cook until flesh easily flakes with a fork for a well-done salmon.
Remove pan from heat and pour lemon garlic sauce over.
Sprinkle with parsley and arrange lemon slices over top.
Serve with skin on or easily remove with a spoon, and serve with more lemon garlic sauce and lemon slices from the pan.