01 - Heat olive oil and butter in a large deep skillet or Dutch oven over medium heat until butter melts and begins to shimmer.
02 - Add diced onions, carrots, and celery to the pan. Sauté for 5-6 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 1-2 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 dried thyme, sage, salt, and black pepper. Bring mixture to a gentle simmer and cook for approximately 5 minutes, stirring occasionally, 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 submerged in the sauce.
08 - Cover and simmer for 10-12 minutes, stirring occasionally to prevent sticking, until gnocchi float freely and are tender when pierced with a fork, and sauce reaches desired consistency.
09 - Remove from heat. Stir in grated Parmesan cheese if using. Garnish with chopped fresh parsley and serve immediately while hot.