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 to the pan. Sauté for 5 to 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 brown the garlic.
04 - Sprinkle flour evenly over the vegetables while stirring constantly. Cook for 1 to 2 minutes to form a roux, allowing the flour taste to cook out.
05 - Gradually whisk in chicken broth and milk or half-and-half, stirring continuously to prevent lumps from forming.
06 - Add dried thyme, sage, salt, and black pepper. Bring mixture to a gentle simmer and cook for approximately 5 minutes until sauce thickens enough to coat the back of a spoon.
07 - Add uncooked gnocchi, cooked chicken, and frozen peas to the simmering sauce. Stir gently to combine, ensuring gnocchi are fully submerged in the liquid.
08 - Cover the pan and reduce heat to maintain a gentle simmer. Cook for 10 to 12 minutes, stirring occasionally, until gnocchi are tender and float to the surface, and sauce reaches desired consistency.
09 - Remove from heat. Stir in grated Parmesan cheese if using, allowing it to melt into the sauce. Garnish with chopped fresh parsley before serving hot.