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 diced onions, carrots, and celery to the hot fat. Sauté for 5-6 minutes, stirring occasionally, until vegetables 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 sautéed vegetables. Stir constantly for 1-2 minutes to cook the flour and form a light roux that will thicken the sauce.
05 - Gradually whisk in chicken broth and milk or half-and-half, stirring continuously to dissolve the roux and prevent any 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 coats the back of a spoon.
07 - Stir in uncooked gnocchi, cooked chicken, and frozen peas. Gently fold to combine, ensuring gnocchi are fully submerged in the creamy sauce.
08 - Cover pot and simmer for 10-12 minutes, stirring occasionally to prevent sticking, until gnocchi are tender and floating, and sauce has reached desired consistency.
09 - Remove from heat. Stir in grated Parmesan cheese if using, allowing residual heat to melt and incorporate the cheese throughout the dish.
10 - Sprinkle with freshly chopped parsley just before serving. Serve hot directly from the pot, spooning generous portions into bowls.