Rinse the mussels and remove beards if necessary, discard any that do not fully close when rinsing (these are dead).
Bring a large pot of water to a boil, add a pinch of salt, and cook linguine until al dente.
Meanwhile, heat butter and oil in a large covered saucepan on medium-high heat. Cook garlic and shallots until fragrant, about 1 minute. Add wine, basil, and red pepper flakes to the pan. Cover and simmer mussels in the white wine sauce until fully cooked and shells open up, about 5 minutes. Stir mussels to coat in the sauce.
When linguine is finished, strain and stir into the mussels and white wine sauce. Now add cream and Parmesan cheese. Simmer for 5-7 minutes, covered, stirring occasionally until the sauce slightly thickens. Serve topped with a pinch of fresh basil, more Parmesan cheese, and 6-7 mussels.