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 begin to soften and onions turn translucent.
03 - Stir in minced garlic and cook for 30 seconds until fragrant, being careful not to burn.
04 - Sprinkle flour evenly over the vegetables, stirring constantly to coat. Cook for 1 to 2 minutes to remove raw flour taste and form a light roux.
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, stirring frequently, until sauce thickens enough to coat the back of a spoon.
07 - Stir in uncooked gnocchi, cooked chicken, and frozen peas. Ensure gnocchi are fully submerged in the sauce for even cooking.
08 - Cover pan and reduce heat to maintain a gentle simmer. Cook for 10 to 12 minutes, stirring occasionally to prevent sticking, until gnocchi are tender and plump.
09 - Remove from heat. Stir in grated Parmesan cheese if using. Garnish with freshly chopped parsley and serve hot.