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 it.
04 - Sprinkle flour evenly over the vegetables, stirring constantly to coat. Cook for 1 to 2 minutes to eliminate raw flour taste and form a thick paste.
05 - Gradually whisk in chicken broth and milk or half-and-half, stirring continuously to prevent lumps. Bring mixture to a gentle simmer.
06 - Add thyme, sage, salt, and pepper. Let sauce simmer for about 5 minutes, stirring occasionally, until it thickens enough to coat the back of a spoon.
07 - Add uncooked gnocchi, cooked chicken, and frozen peas to the sauce. Stir gently to combine, ensuring gnocchi are fully submerged in the liquid.
08 - Cover the pan and simmer for 10 to 12 minutes, stirring occasionally to prevent sticking. Cook until gnocchi are tender and float to the top, and sauce reaches desired creamy consistency.
09 - Remove from heat. Stir in grated Parmesan cheese if using, allowing it to melt into the sauce.
10 - Sprinkle with fresh chopped parsley immediately before serving while still hot.