Preheat the oven to 400 degrees Fahrenheit. Melt 3 tablespoons of butter in a large skillet over medium heat.
Add the onion, carrots, and celery to the skillet. Cook for 5 minutes until the vegetables soften slightly.
Stir in the flour and cook for 1 minute. Slowly whisk in the chicken broth and milk until the mixture thickens.
Add the cooked chicken, frozen peas, thyme, salt, and pepper. Stir everything together and remove the skillet from the heat.
Transfer the filling to a 9-inch baking dish and spread it evenly.
In a mixing bowl, combine 2 cups flour, baking powder, and salt. Cut in the cold butter until the mixture looks like coarse crumbs.
Pour in 3/4 cup milk and stir just until a dough forms. Drop spoonfuls of the dough over the filling.
Bake for 25 minutes until the biscuits turn golden and the filling bubbles at the edges. Let it rest for 5 minutes before serving.