If I'm going to do comfort food, it's got to be done right. In this case, that means making a rock-star solid filling. I avoid using a nonstick skillet here because I want to develop fond (or, as chefs call it, "flavor gold"). Fond is all that amazing business that builds up on the bottom of a pan when you're sauteing something. As long as it stays nutty brown (blackened fond = "flavor kryptonite"), you're primed to then deglaze the pan, which releases all that business and emulsifies it into your sauce. That fond-ation is what makes a sauce that's been cooking for 15 minutes taste like a daylong affair.
Place a large skillet over medium-high heat with the olive oil and butter. Season the chicken liberally with salt and pepper and it to the hot skillet. Saute the chicken until brown, 5 to 6 minutes. Using a slotted spoon, remove the chicken from the pan and set it aside, reserving the drippings in the skillet.
Return the pan to medium heat and add the onion, carrot, celery and garlic. Cook, stirring occasionally, until the vegetables are tender, 4 to 5 minutes. Sprinkle the flour over the vegetables and cook for 1 minute. Deglaze the pan with the wine or sherry (if using), scraping the pan to release any bits stuck to its bottom. Add the chicken stock, milk, mustard and tarragon to the skillet and bring the liquids up to a bubble. Add the reserved chicken to the skillet, season the sauce with salt and pepper and simmer it, stirring occasionally, until thickened, about 5 minutes. Stir in the peas and transfer the filling to a medium baking dish.
While the filling is simmering, unroll the sheet of puff pastry onto a work surface. Cut a few slits through the middle of the dough and brush the surface with a thin layer of the cream. Sprinkle the cheese evenly over the surface.
Carefully place the puff pastry over the baking dish (cheese-side up), pushing the edges gently down into the dish. Place the pot pie onto a baking sheet and bake until the filling is bubbling and the crust is puffed and golden brown, 30 to 35 minutes. Let the pot pie cool for 5 minutes before serving.