Arrange a rack in the middle of the oven and heat the oven to 375°F.
Dice the onion, peel and halve the carrots and celery, and minces the garlic.
Melt 4 tablespoons of butter in a large skillet over medium-high heat and add the onion, carrots, celery, thyme, salt, and pepper. Sauté until softened.
Add flour and garlic, stir to coat, and cook for 1 minute. Slowly pour in chicken broth and milk, scraping the skillet. Bring to a boil.
Reduce heat and simmer until thickened, about 5 minutes. Meanwhile, shred or dice the chicken.
Add chicken and frozen peas to the skillet, stir, season to taste, and transfer to a baking dish.
Open the biscuit can, separate the biscuits, and arrange over the filling.
Bake until biscuits are golden-brown and filling is bubbling, 20-30 minutes. Check and cover if browning too quickly.
Melt remaining 2 tablespoons of butter and brush over the tops of the biscuits.