Preheat the oven to 400°F. Lightly grease a 9x13-inch baking dish.
In a large Dutch oven or deep skillet, melt the butter over medium heat. Add the onion, carrots, and celery. Cook for 7 to 9 minutes, stirring occasionally, until the vegetables begin to soften.
Add the garlic and cook for 30 seconds, just until fragrant.
Sprinkle the flour over the vegetables and stir until the vegetables are evenly coated. Cook for 1 to 2 minutes, stirring constantly, to remove the raw flour taste.
Slowly pour in the chicken stock while stirring, scraping up any browned bits from the bottom of the pan. Add the heavy cream, salt, pepper, and thyme. Bring the mixture to a gentle simmer and cook for 3 to 5 minutes, stirring often, until thickened into a creamy gravy.
Stir in the shredded rotisserie chicken, frozen peas, frozen corn, and parsley. Taste and adjust the seasoning as needed. The filling should taste creamy, savory, and well-seasoned before it goes into the baking dish.
Transfer the hot filling to the prepared baking dish and spread it into an even layer.
Separate the biscuits. Split each biscuit in half horizontally to make thinner biscuit rounds. Arrange the biscuits evenly over the hot chicken filling, leaving a little space between them so they can puff and bake through.
Bake uncovered for 22 to 28 minutes, or until the filling is bubbling and the biscuits are golden brown and cooked through.
Brush the tops of the biscuits with melted butter. Let the casserole rest for 5 to 10 minutes before serving.
