Fill a large pot with water and place over high heat. Bring water to a boil and add a small handful of salt.
Add linguine and bring back up to a boil, stirring occasionally.
Boil linguine until al dente, 6 to 8 minutes.
Transfer to a strainer and drain, reserving 1 cup pasta water. Set linguine and pasta water aside until ready to use.
Add 1 ½ tablespoons unsalted butter and oil into a large skillet and place over medium-high heat.
Season shrimp generously with salt and pepper and sauté in the skillet, in batches, for about 3 minutes on each side.
Remove shrimp from skillet and set aside.
Lower heat to medium and melt remaining butter in skillet.
Add shallots and garlic and sauté for 2 to 3 minutes. Season with salt.
Deglaze the pan with wine and cook until almost fully evaporated, about 4 minutes.
Add reserved pasta water and saffron threads and simmer until liquid has reduced by half.
Stir milk into saffron mixture and simmer. Stir Parmesan into sauce and lightly season with salt and pepper.
Add shrimp back into the skillet and simmer in sauce for 1 to 2 minutes.
Add linguine, to skillet and toss together with shrimp and garlic-saffron cream.
Top pasta with more grated Parmesan and fresh basil leaves. Serve.