Bring a large pot of salted water to a boil. Add pasta and cook according to package directions until al dente. Reserve 1 cup of pasta water before draining.

Pat salmon fillets dry and season with salt and pepper. Heat olive oil and 1 tbsp of butter in a large skillet over medium-high heat. Sear salmon for 4-5 minutes per side, until cooked through. Remove from skillet and set aside.

Reduce heat to medium. Add the remaining 2 tbsp of butter and the minced garlic to the same skillet. Sauté for 30 seconds until fragrant. Pour in the heavy cream and simmer for 2-3 minutes to thicken. Stir in the Parmesan cheese until melted.

Add the fresh spinach to the sauce and stir until wilted. Flake the cooked salmon into large chunks and add it to the sauce. Add the drained pasta and toss to combine. If needed, add reserved pasta water to reach desired sauce consistency. Season with salt, pepper, and red pepper flakes.

Serve immediately, garnished with extra Parmesan cheese if desired.