Cook ½ lb spaghetti noodles in a large pot of salted water according to package instructions until al dente. Drain and set aside.
Season the ½ lb salmon fillet with salt and black pepper. Slice the fish in half if it's too big to fit in the pan.Add 1 tablespoon of salted butter and 1 tablespoon of olive oil to a large pan over medium high heat.
Once the pan is hot, place the fish skin-side up and sear for 2-4 minutes. Flip over and sear another 2-4 minutes, or until fish is opaque and cooked.
Remove the salmon and add to a plate. Use two forks to flake the fish into large chunks and remove it from the skin. Set aside for now.
Add another 1 tablespoon of olive oil to the pan over medium heat along with 3 minced garlic cloves. Cook for 1 minute. Pour in the ½ cup chicken broth. Scrape down the bottom of the pan using a spatula to remove any baked-on bits. Let the broth cook for 4-5 minutes to let it reduce.
Now add 1 cup heavy cream, 2 cups baby spinach, 1 cup grape tomatoes, ½ cup grated parmesan, 2 tablespoon lemon zest, and ½ teaspoon salt. Mix together. Cook for 1 minute to soften the spinach.
Remove the pan from the heat. Add the cooked pasta and flaked salmon.
Toss everything together. Sprinkle with more grated parmesan if desired.
Scoop on to serving plates. Drizzle with freshly squeezed lemon juice and a bit of olive oil. Season with salt and pepper.