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 pot. 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 brown or burn it.
04 - Sprinkle flour evenly over the vegetables, stirring constantly to coat everything. Cook for 1 to 2 minutes to eliminate raw flour taste and form a smooth roux base.
05 - Gradually whisk in chicken broth and milk or half-and-half, stirring continuously to prevent lumps from forming. Add thyme, sage, salt, and black pepper.
06 - Bring mixture to a gentle simmer and cook for about 5 minutes, stirring frequently, until sauce noticeably thickens and coats the back of a spoon.
07 - Stir in uncooked gnocchi, shredded chicken, and frozen peas. Gently fold everything together, ensuring gnocchi are fully submerged in the liquid.
08 - Cover pot and reduce heat to maintain a gentle simmer. Cook for 10 to 12 minutes, stirring occasionally to prevent sticking, until gnocchi are tender and sauce has reached a creamy consistency.
09 - Remove pot from heat. Stir in grated Parmesan until melted and incorporated throughout the sauce.
10 - Ladle hot portions into bowls and sprinkle generously with chopped fresh parsley just before serving.