Arrange a rack in the middle of the oven and heat the oven to 375°F.

Dice the yellow onion. Peel and halve the carrots and celery, then cut into 1/2-inch wide pieces. Mince the garlic. Pick the leaves from the thyme and coarsely chop.

Melt 4 tablespoons of butter in a large skillet over medium-high heat. Add the onion, carrots, celery, thyme, salt, and black pepper. Sauté until the vegetables begin to soften and brown, 5 to 7 minutes.

Add the flour and garlic. Stir to coat the vegetables and cook until fragrant, about 30 seconds to 1 minute. Slowly pour in the chicken broth and milk, scraping the bottom of the skillet. Bring to a boil.

Reduce the heat and maintain a simmer until thickened, about 4 to 5 minutes. Meanwhile, shred or dice the cooked chicken into bite-sized pieces.

Add the chicken and frozen peas to the skillet, stirring to combine. Taste and adjust seasoning if needed. Transfer to a 9x13-inch baking dish and spread into an even layer.

Open the can of biscuits, separate the biscuits, and arrange them in a single layer over the filling.

Bake until the biscuits are golden-brown and cooked through, and the filling is bubbling, 20 to 30 minutes, checking after 15 minutes for browning.

Melt the remaining 2 tablespoons of butter and brush over the tops of the biscuits.