Start by melting butter in a Dutch oven over medium-high heat. Add carrots, mushrooms, celery, and shallot, then season with salt and pepper. Cook until the mushrooms release their liquid and everything softens.
Stir in the garlic, poultry seasoning, and a pinch of dried thyme. Cook for just a minute until the garlic becomes fragrant. This deepens the flavor base before adding liquids.
Sprinkle the flour over the warm vegetables and stir until combined. Slowly pour in the chicken broth while stirring to avoid lumps. Add the milk and bring the mixture to a gentle simmer.
Stir in the gnocchi and keep the pot simmering. Cook for 5–6 minutes, stirring occasionally, until the gnocchi becomes tender and slightly plump.
Add the shredded chicken and frozen peas, stirring them into the creamy mixture. Taste and adjust seasoning before serving.