Cook oil, garlic, and crushed red pepper in a medium saucepan over medium-high heat, stirring constantly, until fragrant and sizzling, about 2 minutes.
Stir in wine; cook, stirring once, until reduced by half, about 2 minutes.
Stir in broth, salt, and pepper. Cover and bring to a boil over high heat.
Add pasta; cook, stirring occasionally, until al dente, about 6 minutes.
Remove from heat. Add chicken and spinach; stir until the spinach is wilted, about 1 minute.
Stir in lemon zest and lemon juice.
Divide the soup among 4 shallow bowls; sprinkle with Parmesan.