Bring a large pot of salted water to a boil. Cook pasta until al dente, reserving ⅛ cup pasta water before draining.

In a large sauté pan over medium-high heat, add 1 tablespoon olive oil and 1 tablespoon butter.

Add shrimp and half the garlic. Season lightly with salt. Cook until just opaque (about 2 minutes).

Remove shrimp and set aside.

In the same pan, add 1 tablespoon olive oil and ½ tablespoon butter over medium heat.

Add cherry tomatoes and remaining garlic. Cook until tomatoes burst and soften (5–7 minutes).

Deglaze with wine, raise heat, and boil until reduced by half.

Add chicken stock, reduce by half again.

Add heavy cream, reduce until slightly thickened.

Add shrimp and arugula to sauce. Stir to combine and wilt arugula.

Season with 2 teaspoon salt and 1 teaspoon pepper. Add pasta water.

Add drained pasta directly to the sauce. Toss to combine and coat evenly.

Turn off heat, sprinkle with pecorino, adjust seasoning, and serve immediately.