HEAT the butter and oil in a large oven-proof pan (cast-iron or stainless steel are great here) over medium heat. Add the chicken and ½ teaspoon of kosher salt and cook it for about 3-4 minutes, until it's browned on at least one side, but still pink in the middle.
ADD the sliced onions to the pan and cook them until they become translucent, another 3-4 minutes. Then add the frozen mixed vegetables with an additional ½ tsp kosher salt, and cook for an additional 3-5 minutes, until the vegetables are fully defrosted.
SPRINKLE the flour over the top of the vegetables and stir until the flour is all dissolved into the pan. Cook it for an additional 1-2 minutes to cook out the flour taste. Then add the milk slowly and continue to stir to ensure there aren't any lumps. Season the pie filling with salt and lots of pepper and turn off the heat.
PREHEAT the oven to 425F
COMBINE flour, baking powder, baking soda, and salt in a medium bowl and toss to combine. Add the butter pieces and work it with your fingers until the butter pieces are no bigger than peas.
ADD the milk and stir gently until it just starts to come together as a dough. If you squeeze a piece in your hand it should hold together well but it doesn't have to be a solid mass.
TURN OUT the dough onto a floured surface and pat it into a uniform shape that is about ¾ inch thick.
CUT circles (or any other shape) out of the dough to form individual biscuits, about 2 inches in diameter (or across). You'll probably get have to reshape the scraps once to get about 10 biscuits.
ARRANGE the biscuits on the prepared filling and stick the whole pan in the oven to bake for 15-18 minutes, until the biscuits are browned on top. Remove from the oven, sprinkle with the parsley (if using), and serve while it's still hot.
This recipe can be made even easier if you use a store bought biscuit dough. No one will judge you for it, and you can always lie and say you made the biscuits from scratch if they do. I won't give away your secret.
This pie is best served hot, although it reheats well and you can keep the leftovers for about a week in the fridge.