01 - Heat olive oil and butter in a large deep skillet or Dutch oven over medium heat until butter melts and begins to sizzle.
02 - Add onions, carrots, and celery to the skillet. Sauté for 5 to 6 minutes, stirring occasionally, until vegetables are softened and onions are 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 to 2 minutes to cook off the 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. Bring the mixture to a gentle simmer.
06 - Add dried thyme, sage, salt, and black pepper. Allow the sauce to simmer gently for about 5 minutes, stirring occasionally, until it thickens noticeably and coats the back of a spoon.
07 - Stir in uncooked gnocchi, cooked chicken, and frozen peas. Gently fold everything together, ensuring the gnocchi are fully submerged in the sauce.
08 - Cover the skillet 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 thickened to a creamy consistency.
09 - Remove from heat. Stir in grated Parmesan cheese if using, allowing it to melt into the sauce for extra richness.
10 - Sprinkle freshly chopped parsley over the top as a garnish. Serve hot directly from the skillet or transfer to serving bowls.