Bring a large pot of salted water to a boil, then add linguine and cook until al dente according to package directions. Before draining, reserve 1 cup of the pasta water.

Pat the salmon fillets dry and season with salt and pepper. Heat 1 tablespoon of olive oil in a large skillet over medium-high heat. Cook the salmon for 4-6 minutes per side until cooked through and flaky, then remove, cool slightly, and flake into bite-sized pieces.

Reduce heat to medium in the same skillet. Add the remaining olive oil if needed, and minced garlic; sauté for 30 seconds until fragrant, adding red pepper flakes now if desired. Pour in heavy cream and broth, bringing to a gentle simmer, then stir in Parmesan cheese until melted and the sauce slightly thickens.

Add the drained linguine to the skillet and toss to coat. If the sauce is too thick, add reserved pasta water until the desired consistency is reached before gently folding in the flaked salmon.

Stir in fresh lemon juice and adjust seasoning with salt and pepper as needed. Garnish generously with fresh parsley and dill, then serve immediately.