Cook linguine with tablespoon salt to 1 minute short of package time. Reserve 1 cup pasta water; drain.
Meanwhile, pat shrimp dry; season with their salt and pepper. Heat oil and 1 tablespoon butter in large skillet; cook shrimp 1 to 2 minutes per side until opaque and at least 145°F. Plate.
Lower heat; melt remaining butter. Stir garlic 30 seconds; add cream, broth, and zest. Gently simmer 2 minutes.
Add linguine and 1/4 cup reserved water; toss 1 to 2 minutes until tender. Remove from heat; gradually add Parmesan.
Return shrimp; toss just to warm. Add pasta water if needed; serve.