Cook the Salmon: Heat olive oil in a large skillet over medium-high heat. Season the salmon fillets with salt and pepper. Add the salmon to the skillet, skin-side down, and cook for 4–5 minutes per side, until golden and cooked through (internal temperature of 145°F/63°C). Remove the salmon from the skillet and set aside.
Make the Creamy Spinach Sauce: In the same skillet, melt butter over medium heat. Add the minced garlic and sauté for 1–2 minutes until fragrant.
Add Spinach and Liquids: Add the fresh spinach to the skillet and sauté until wilted, about 2–3 minutes. Pour in the heavy cream and chicken broth, stirring to combine.
Season the Sauce: Stir in the lemon juice, dill, salt, and pepper. Let the sauce simmer for 3–4 minutes, allowing it to thicken slightly.
Finish the Sauce: Stir in the Parmesan cheese, allowing it to melt into the sauce and make it creamy.
Combine Salmon and Sauce: Return the cooked salmon to the skillet and spoon the creamy spinach sauce over the top. Let it simmer for an additional 2–3 minutes to heat everything through.
Serve: Garnish with fresh parsley if desired and serve immediately, paired with rice, pasta, or roasted vegetables.