For the Biscuits:

1. Whisk the flour, baking powder, baking soda, sugar, and salt together in a mixing bowl. Stir in the grated cheese until it is coated in the flour.
2. Melt the butter in a heat proof mixing cup in the microwave. Once it has cooled slightly, whisk in the buttermilk. If the mixture looks curdled, this is OK! It will incorporate once mixed with the dry ingredients.
3. Pour the wet ingredients into the dry and stir until the mixture comes together and is no longer crumbly.
4. Set aside.

For the Filling:

1. Preheat the oven to 400°F.
2. On the stove, heat a large oven-safe skillet or pan over medium heat. Add the oil and butter.
3. Once the butter is melted, add the onion, carrots, celery, and a pinch of salt. Cook, stirring frequently, until the veggies begin to soften, about 5 minutes.
4. Stir in the thyme and garlic and cook for another minute. Sprinkle over the flour and stir to combine.
5. Slowly whisk in the milk and chicken broth and stir over medium heat until the mixture begins to thicken, about 5 minutes. Stir in the cheese until it is melted. Season the sauce, to taste, with salt and pepper.
6. Once the sauce is creamy and slightly thickened, stir in the shredded chicken, broccoli, and peas. Remove the pan from the heat.

To Assemble:

1. Dollop ¼-cup scoops of the biscuit mixture all over the top of the filling. Don’t worry about making them perfect – they’re going to puff up substantially and cover up any imperfections in the oven. You want them to be roughly the same size so they bake evenly.
2. Brush the tops of the biscuits with a little buttermilk and place the pan, uncovered, in the oven for 25-30 minutes or until the biscuits are golden and puffed and the filling is bubbly around the edges.