Bring a large pot of salted water to a boil. Cook the pasta according to package directions until al dente. Reserve about 1/2 cup of pasta water before draining.

While the water heats, pat the salmon dry. Season both sides with salt and pepper, then lightly coat with flour, shaking off any excess.

In a large skillet over medium-high heat, warm the olive oil and melt the butter. Once shimmering, add the salmon and cook for 2 minutes per side, until golden. Transfer the salmon to a plate. If the skin is still on and you prefer it skinless, peel it off now.

Reduce the heat to medium. Add the chicken broth, minced garlic, and lemon juice to the skillet. Simmer for about 1 minute, scraping up any browned bits from the bottom.

Return the salmon to the pan, breaking it into bite-sized pieces with a spoon. Cook for 2–3 minutes more, until the salmon is cooked through and the sauce has slightly thickened.

Drain the pasta and add it to the skillet. Toss everything together, adding a splash of reserved pasta water if the sauce seems too thick. Serve immediately, topped with parsley, red pepper flakes, and Parmesan if desired.