01 - Heat olive oil and butter in a large deep skillet or Dutch oven over medium heat until butter melts.
02 - Add onions, carrots, and celery. Cook for 5–6 minutes, stirring occasionally, until vegetables begin to soften.
03 - Stir in garlic and cook for 30 seconds until fragrant, being careful not to burn.
04 - Sprinkle flour evenly over vegetables while stirring constantly. Cook for 1–2 minutes to eliminate raw flour taste and form a thick paste.
05 - Gradually whisk in chicken broth, followed by milk or half-and-half. Stir 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 until sauce coats the back of a spoon.
07 - Stir in uncooked gnocchi, cooked chicken, and frozen peas. Ensure gnocchi are fully submerged in the liquid.
08 - Cover pot and simmer for 10–12 minutes, stirring occasionally to prevent sticking, until gnocchi are tender and sauce has thickened.
09 - Remove from heat. Stir in Parmesan cheese until melted and incorporated. Garnish with fresh parsley and serve immediately.