Melt the butter in a large pot over medium-high heat. Add the minced garlic and chopped onion, cooking for 2 minutes. Stir in the thyme, chopped carrot, and celery, and cook for an additional 3 minutes or until the carrots begin to soften. Pour in the white wine, stirring and scraping the bottom of the pot to deglaze, cooking until the wine has mostly evaporated, about 1.5 minutes. Sprinkle in the flour, stirring continuously for 1 minute to create a roux. Gradually add half of the reserved poaching liquid, stirring until the flour is fully incorporated and the mixture thickens. Pour in the remaining milk-broth mixture, followed by the grated parmesan and black pepper. Stir to combine. Add the shredded chicken and frozen peas, mixing well. Once the mixture starts to steam, cook for 3 more minutes, stirring regularly to thicken the sauce. Remove from heat and spoon the mixture into oven-proof pots, either 4 large or 6 small ones. Let the pots cool in the fridge for at least 30 minutes.