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 to 6 minutes, stirring occasionally, until vegetables begin to soften and onions turn 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, stirring constantly to coat. Cook for 1 to 2 minutes to form a roux and eliminate 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 about 5 minutes, stirring occasionally, 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 and simmer for 10 to 12 minutes, stirring occasionally to prevent sticking, until gnocchi float to the surface and are tender when tasted.
09 - Remove from heat. Stir in grated Parmesan cheese if using. Garnish generously with fresh chopped parsley and serve hot.