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 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 while stirring constantly. Cook for 1 to 2 minutes to form a roux, allowing the flour taste to cook out and mixture to bubble gently.
05 - Gradually whisk in chicken broth and milk or half-and-half, stirring continuously to prevent lumps from forming. Ensure the roux is fully dissolved into the liquid.
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 noticeably thickens and coats the back of a spoon.
07 - Stir in uncooked gnocchi, shredded chicken, and frozen peas. Gently fold ingredients together to ensure gnocchi are fully submerged in the sauce.
08 - Cover pot and reduce heat to maintain a gentle simmer. Cook for 10 to 12 minutes, stirring every few minutes to prevent sticking, until gnocchi float to the surface and are tender when tasted.
09 - Remove pot from heat source. Stir in grated Parmesan cheese until melted and incorporated throughout the creamy sauce.
10 - Sprinkle chopped fresh parsley over the top immediately before serving. Serve hot directly from the pot, allowing guests to spoon into bowls.