How to Make It
Make filling: Place chicken in a medium pot and pour in broth. Bring to a simmer over high heat. Reduce heat to medium, cover and cook until chicken is no longer pink, about 10 minutes. Remove meat from broth and set aside to cool.
Add a pinch of salt to broth in pot and return to a boil. Add potatoes and carrots and cook until vegetables are just tender, about 7 minutes. Transfer vegetables to a large bowl, reserving broth in a heatproof bowl. Wipe out pot.
Melt 2 Tbsp. butter with oil in a large skillet over medium-high heat. Add mushrooms and cook, stirring often, until they begin to give off liquid and start to brown, about 7 minutes. Season with salt and pepper. Using a slotted spoon, transfer mushrooms to bowl with other vegetables, leaving oil in skillet.
In same pot used for vegetables, melt remaining 2 Tbsp. butter over medium-high heat. Whisk in flour, then cook, whisking constantly, until mixture is blond in color, about 2 minutes. Pour in reserved chicken cooking liquid, increase heat to high and bring to a boil. Whisk until thickened and smooth. Pour in cream and remove from heat. Season with salt and pepper. Cut chicken into 1-inch cubes. Combine chicken, mushrooms, carrots, potatoes and peas in a large bowl. Stir in sauce; season with salt and pepper. Cover and chill until ready to make topping.
Preheat oven to 375°F; lightly butter 6 1 1/2- to 2-cup-capacity ovenproof ramekins. Divide filling evenly among ramekins.
Make topping: Pulse flour, baking powder and salt in a food processor to mix. Add butter and process in several quick pulses until mixture resembles coarse meal. Add buttermilk all at once. Pulse several times to mix, then turn out onto a floured work surface. Gather dough into a ball and, using fingers, pat out to a thickness of about 1/2 inch, dusting with more flour if dough is sticky. Use a round cutter to cut out small biscuits. Gather and reroll scraps until all dough has been used. Place 2 to 3 biscuits on top of each ramekin and brush biscuits with buttermilk. Bake until filling is bubbly and biscuits are lightly browned, 30 to 35 minutes. Serve hot.