In a large, deep, nonstick skillet over medium heat, cook bacon, turning frequently, until crisp and fat is rendered, 12 to 15 minutes. Transfer to paper towels to drain. Let cool slightly, then transfer to a cutting board and crumble into small pieces.

Pour rendered bacon fat into a cup. Return 2 Tbsp. fat to skillet; reserve remaining fat for another use. Add chicken and cook over medium heat, turning occasionally, until cooked through and an instant-read thermometer inserted into a piece registers 165°, about 5 minutes. Add garlic and cook, stirring, until fragrant, about 1 minute more. Stir in Italian seasoning.

Add gnocchi, cream, and broth. Return most of bacon to pan; reserve remaining bacon for serving. Stir to combine, then reduce heat to medium-low. Cover with a tight-fitting lid and cook 5 minutes. Uncover and continue to cook, stirring occasionally, until sauce is reduced and slightly thickened, about 5 minutes more; season with salt and pepper, if needed. Add cheese and stir until melted and creamy.

Divide gnocchi mixture among bowls. Top with reserved bacon and chives (if using).