01 - Heat olive oil and butter in a large deep skillet or Dutch oven over medium heat until butter melts and mixture shimmers.
02 - Add onions, carrots, and celery. Sauté for 5-6 minutes, stirring occasionally, until vegetables soften and onions become translucent.
03 - Stir in minced garlic and cook for 30 seconds until fragrant, being careful not to burn it.
04 - Sprinkle flour evenly over the vegetables while stirring constantly. Cook for 1-2 minutes to form a roux, eliminating raw flour taste.
05 - Gradually whisk in chicken broth and milk or half-and-half, stirring continuously to prevent lumps from forming.
06 - Add thyme, sage, salt, and pepper. Bring mixture to a gentle simmer and cook for approximately 5 minutes, stirring frequently, until sauce thickens enough to coat the back of a spoon.
07 - Stir in uncooked gnocchi, cooked chicken, and frozen peas. Gently fold to combine, ensuring gnocchi are fully submerged in the sauce.
08 - Cover and reduce heat to maintain a gentle simmer. Cook for 10-12 minutes, stirring occasionally to prevent sticking, until gnocchi are tender and floating, and sauce has reached a creamy consistency.
09 - Remove from heat. Stir in grated Parmesan if desired. Garnish generously with chopped fresh parsley before serving hot.