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 hot 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.
04 - Sprinkle flour evenly over the vegetables, stirring constantly to coat. Cook for 1–2 minutes to eliminate raw flour taste and form a thick paste.
05 - Gradually whisk in chicken broth and milk or half-and-half, about 1/2 cup at a time, stirring vigorously to prevent lumps from forming.
06 - Add thyme, sage, salt, and black pepper. Bring mixture to a gentle simmer and cook for approximately 5 minutes, stirring frequently, until sauce noticeably thickens enough to coat the back of a spoon.
07 - Add uncooked gnocchi, shredded chicken, and frozen peas to the sauce. Stir gently to distribute evenly and ensure gnocchi are fully submerged in liquid.
08 - Cover pot and reduce heat to maintain a gentle simmer. Cook for 10–12 minutes, stirring occasionally to prevent sticking, until gnocchi float to the surface and feel tender when tested.
09 - Remove pot from heat. Stir in grated Parmesan cheese until melted and incorporated into the creamy sauce.
10 - Transfer to serving bowls and sprinkle generously with chopped fresh parsley. Serve immediately while hot and creamy.