Preheat the oven to 400°F. Melt 3 tablespoons of butter in an oven-safe skillet over medium heat. Add the diced onion, sliced carrots, and sliced celery. Cook for 5 minutes until the vegetables begin to soften.

Stir in the flour and cook for 1 minute. Slowly whisk in the chicken broth and milk until the mixture thickens into a sauce. Add the shredded chicken, frozen peas, thyme, salt, and pepper. Simmer for 2 minutes, then remove from heat.

In a mixing bowl, combine 1 1/2 cups flour, baking powder, and 1/2 teaspoon salt. Cut in the 4 tablespoons cold cubed butter until the mixture looks like coarse crumbs. Stir in 1/2 cup milk until a soft dough forms.

Drop spoonfuls of the biscuit dough evenly over the surface of the chicken filling. Place the skillet in the oven and bake for 20 minutes until the biscuits are golden and the filling bubbles around the edges.