1. Prepare an ice-water bath; set aside. In a medium saucepan of boiling salted water, cook fava beans 1 minute. Drain and immediately transfer beans to ice-water bath until cool. Peel beans; set aside.

2. Melt butter in an 11 1/2-inch paella pan over medium-high heat. Add shrimp and cook, stirring, until bright red, about 1 minute; remove shrimp from pan and set aside. Add pasta to paella pan and stir until well coated with butter. Cook until evenly browned and toasted, 2 to 3 minutes. Add onion and garlic; cook, stirring, for 1 minute. Add stock, 1/2 cup at a time, stirring, and allowing pasta to absorb stock after each addition before adding more. Cook until tender, about 10 minutes.

3. Add pepper paste, roasted peppers, olives, reserved fava beans, and basil; stir to combine. Add shrimp and cook until heated through, about 3 minutes. Sprinkle with pecorino Romano. Divide evenly between 2 plates and drizzle with olive oil. Serve immediately.