Bring a large pot of salted water to a boil over medium-high heat. Add the pasta and cook, stirring occasionally, just until they are al dente. This should take about 8 minutes but make sure to check the package directions for the exact time. Before you drain the pasta, make sure to reserve ½ cup of the starchy pasta water for later.
Heat the olive oil in a large, high-sided skillet set over medium heat.
Pat the salmon dry on both sides with a paper towel and season it lightly with salt and pepper. When the oil is hot, add the salmon skin-side down to the pan. Cook for 3–4 minutes, or until the salmon is golden brown and releases naturally from the pan. If you have to pull the salmon off the pan, it’s not ready to flip yet!
After you’ve flipped the salmon, add the butter, minced garlic, minced thyme, and lemon juice to the pan. Cook the salmon for an additional 2–3 minutes, or until it is light pink and flaky. Baste the salmon with butter a few times and then remove it to a plate.
Add the flour to the pan with the garlic butter and whisk it into the butter and oil to create a roux.
Next, add the chicken stock slowly, whisking constantly. Cook the sauce over medium heat for 1–2 minutes, or just until it begins to bubble around the edges.
Add ¼ cup of the reserved pasta water to the sauce along with the heavy cream and Parmesan cheese. Whisk to combine. Season the sauce with salt and pepper to taste.
Add the cooked pasta to the sauce and toss to combine. If the sauce becomes too dry, add the remaining ¼ cup of reserved pasta water.
Next, cut the salmon into bite-sized pieces and add it to the pasta. Add the spinach and stir to combine– the spinach will wilt quickly.
Serve immediately, garnished with chopped parsley.