Adjust the oven rack to the center position and preheat oven to 400°F.
In a large pot or Dutch oven set over medium heat, melt the butter. Add the garlic and carrots and cook until tender, stirring occasionally, for about 5 to 7 minutes. Mix in the chicken. Sprinkle the flour over the mixture and stir to combine. Cook for 2 minutes, stirring constantly. Pour in the chicken broth, then add the half-and-half and the wine, if using. Stir to combine, scraping the bottom of the pot. Increase the heat and bring to a simmer, stirring occasionally. Continue to stir occasionally while the mixture simmers for a minute or two, or until thickened. Stir in the thyme, garlic powder, onion powder, salt, and pepper. Adjust the seasonings to taste. Stir in the cubed potato and the frozen green peas, then remove the pot from the heat.
Pour the mixture into a deep 8-by-8-inch baking dish (or a baking dish of a similar size/capacity). Roll out the pie crust so that it's about one inch larger than your baking dish. Place the crust on top to the pot pie mixture and press the edges into the size of the dish. Cut vents in the top of the crust.
Bake for 30 to 35 minutes or until golden and bubbly. Allow to cool for 10 minutes before serving.