Bring a large pot of salted water to a boil. Add the linguine and cook according to the package instructions until al dente. Drain the pasta, reserving ½ cup of the pasta water, and set aside.
In a large skillet, heat 2 tablespoons of olive oil over medium-high heat. Add the shrimp and season with kosher salt and black pepper. Cook for 2-3 minutes on each side until they are pink and opaque. Remove the shrimp from the skillet and set aside.
In the same skillet, add 3 tablespoons of olive oil and warm over medium-high heat. Add the butter, asparagus and scallions (white and pale green parts only). Cook for 3 minutes. Add the minced garlic and red pepper flakes, and cook for another minute until fragrant.
Add the dry white wine and scrape up any brown bits from the bottom of the skillet. Reduce the wine by half, about 2-3 minutes. Stir in the cherry tomatoes and cook until they start to soften, about 2 minutes. Add the heavy whipping cream, lower the heat to medium and bring to a simmer.
Add the cooked shrimp to the skillet. Drain the linguine and reserve ½ cup of the water. Add the drained linguine to the skillet and toss to combine. If the sauce is too thick, add a small amount of the reserved pasta water a little at a time until you reach your desired consistency. Go slowly, the pasta water. Stir in the chopped fresh basil and 2 tablespoons of butter. Stir to combine.
Garnish with the scallion tops and a little bit of freshly ground black pepper. Serve immediately.