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 onions, carrots, and celery to the hot fat. 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 let it brown or burn.
04 - Sprinkle flour evenly over the vegetables, stirring constantly to coat. Cook for 1 to 2 minutes to eliminate raw flour taste and form a light roux base.
05 - Gradually whisk in chicken broth and milk or half-and-half, stirring continuously to prevent lumps from forming. Continue whisking until mixture is smooth.
06 - Add dried thyme, sage, salt, and black pepper. Bring sauce to a gentle simmer and cook for about 5 minutes, stirring occasionally, until sauce thickens enough to coat 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 liquid.
08 - Cover pot with lid and simmer for 10 to 12 minutes, stirring occasionally to prevent sticking. Cook until gnocchi are tender and float to the surface, and sauce has reached desired creamy consistency.
09 - Remove from heat. Stir in grated Parmesan cheese if using, allowing residual heat to melt and incorporate.
10 - Transfer to serving bowls and garnish generously with chopped fresh parsley. Serve immediately while hot.