Cook linguine in water with the tablespoon salt until 1 minute short of the package time. Reserve 1 cup pasta water before draining.
Meanwhile, pat shrimp dry and season with their salt and pepper. Heat oil and 1 tablespoon butter in a large skillet and cook shrimp for 1 to 2 minutes per side until opaque and at least 145°F; transfer to a plate.
Lower heat and melt remaining butter in the skillet. Stir garlic for 30 seconds, then add cream, broth and lemon zest and simmer gently for 2 minutes.
Add linguine and 1/4 cup reserved pasta water and toss for 1 to 2 minutes until tender. Remove from heat and stir in Parmesan gradually.
Return shrimp and toss just to warm them in the sauce, adding more pasta water if needed before serving.