Cut off broccoli florets and save for another use. Cut woody ends off the stems. Shave large knots off the stems using a vegetable peeler so that they are as uniform as possible. Cut into noodles using the smallest blade on a spiralizer.

Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Add broccoli noodles, salt, and pepper; toss for 3 minutes. Remove from heat and set aside.

Heat remaining olive oil and garlic in a separate skillet over medium heat. Cook for 1 minute. Add shrimp and cook until opaque, about 3 minutes per side. Transfer shrimp to a bowl.

Add wine, lemon juice, butter, basil, chives, and red pepper to the skillet. Whisk over medium heat for 3 minutes. Return shrimp to the skillet and toss to coat.

Spoon broccoli noodles into 4 serving bowls. Top with shrimp mixture.