Heat 2 teaspoon oil in an oven safe Dutch oven over medium heat. Add onions and carrots, and cook, stirring occasionally, for about 10 minutes. Remove from pan, and set aside. Heat the remaining 1 teaspoon oil in the Dutch oven over medium-high heat, and add the mushrooms. Cook, stirring often, for about 7-10 minutes, until mushrooms release their juices. Add garlic, and stir constantly for about 30 seconds.
Return onions and carrots back to the pan. Add 2 cups of chicken broth to the Dutch oven, and bring to a boil. Reduce heat to a simmer.
Whisk together the remaining ½ cup chicken broth and the cornstarch. Make sure there are no lumps and the mixture is completely smooth. Slowly stir the cornstarch mixture into the pan. Continue stirring until the mixture has thickened. Stir in the chicken, peas, salt, pepper, and thyme. Remove from heat, and stir in the yogurt. Be sure to mix well.
The mixture can be transferred to a baking dish if desired. Use a 3.5 to 4 quart baking dish.
For the biscuit topping
Meanwhile, whisk together the flours, sugar, baking powder, baking soda, salt, and thyme. With a pastry blender or two knives, cut in the butter until mixture is crumbly. Add buttermilk and oil. Stir until just combined. Drop biscuit topping evenly onto the filling.
If you're using a baking dish: place on a sturdy, rimmed baking sheet. Bake the pot pie for 30 minutes or until biscuit topping is golden and the filling is bubbling. Remove from oven and let sit for 5-10 minutes before serving.
*Based off a recipe from Eating Well*I prepared the filling in advance, and it didn't work very well. Once the mixture cooled, it thinned out and wasn't thick like a regular pot pie filling usually is. Either way, the pot pie was still delicious. Just thought I'd warn you.