Preheat the oven to 400°F (200°C).
In a large oven-safe skillet or Dutch oven, melt 2 tablespoons of unsalted butter over medium heat.
Add the diced onion, carrots, and celery to the skillet and sauté for 5-7 minutes until the vegetables are softened.
Add the minced garlic and cook for 1 minute until fragrant.
Sprinkle 1/4 cup all-purpose flour over the vegetables and stir well to coat. Cook for 2 minutes to remove the raw flour taste.
Gradually whisk in 2 cups chicken broth and 1 cup whole milk, stirring constantly to avoid lumps.
Bring the mixture to a simmer and cook until it thickens, about 3-5 minutes.
Stir in the shredded cooked chicken, 1 cup frozen peas, 1 teaspoon dried thyme, 1/2 teaspoon salt, and 1/4 teaspoon black pepper. Mix well and remove from heat.
In a medium bowl, combine 1 1/2 cups all-purpose flour, 2 teaspoons baking powder, and 1/2 teaspoon salt.
Cut in 6 tablespoons cold unsalted butter using a pastry cutter or your fingers until the mixture resembles coarse crumbs.
Pour in 3/4 cup whole milk and stir just until combined to form a soft dough.
Drop spoonfuls of the biscuit dough evenly over the chicken mixture in the skillet.
Place the skillet in the preheated oven and bake for 20-25 minutes until the biscuits are golden brown and cooked through.
Remove from oven and let cool for 5 minutes before serving.