Adjust the oven rack to the center position and preheat oven to 375°F.
Pour the olive oil into a large pot or Dutch oven set over medium to medium-high heat. 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, 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 4 1/2- to 5-quart baking dish (or a baking dish of a similar size/capacity); set aside.
In a large food processor, pulse together the flour, baking powder, and salt until combined. Add the butter and pulse about ten (1-second) pulses until the mixture resembles coarse sand with some small pebbles mixed in. Add the buttermilk and pulse until all ingredients are fully incorporated. The dough should be very sticky but not runny.
Use a large spoon to scoop and dollop blobs of biscuit dough onto the top of the chicken pot pie filling...I made four rows of three biscuits for a total of 12 biscuits. Carefully brush the top of each biscuit with some half-and-half.
Bake for 35 to 40 minutes until the tops of the biscuits are golden and the filling is bubbly. Allow to cool for 5 to 10 minutes before serving.
If you prefer to make the biscuits without using a food processor, simply whisk the dry ingredients together in a large bowl. Use a pastry cutter or two forks to cut the butter into the flour until the mixture resembles coarse crumbs. Stir in the buttermilk until combined before proceeding with the recipe.