01 - Heat olive oil and butter in a large deep skillet or Dutch oven over medium heat until butter melts and bubbles gently.
02 - Add diced onions, carrots, and celery to the skillet. Sauté for 5–6 minutes, stirring occasionally, until vegetables begin to soften and onions become translucent.
03 - Stir in minced garlic and cook for 30 seconds until fragrant, being careful not to brown or burn the garlic.
04 - Sprinkle flour evenly over the vegetables, stirring constantly to coat everything. Cook for 1–2 minutes to eliminate raw flour taste and form a thick paste.
05 - Gradually whisk in the chicken broth followed by the milk or half-and-half, stirring continuously to break up any lumps and create a smooth base.
06 - Add dried thyme, sage, salt, and black pepper. Bring the mixture to a gentle simmer and cook for approximately 5 minutes, stirring occasionally, until sauce noticeably thickens and coats the back of a spoon.
07 - Stir in uncooked gnocchi, cooked chicken, and frozen peas. Gently fold to combine, ensuring the gnocchi are fully submerged in the liquid.
08 - Cover the skillet and reduce heat to maintain a gentle simmer. Cook for 10–12 minutes, stirring occasionally to prevent sticking, until gnocchi are tender and floating, and the sauce has reached a creamy consistency.
09 - Remove from heat. Stir in grated Parmesan cheese if using, allowing it to melt into the sauce. Garnish with chopped fresh parsley and serve hot.