Season the chicken generously with salt and pepper and arrange in a single layer in a heavy flameproof pan, such as a large skillet or small roasting pan.
Roast the chicken until tender and the juices run clear, or it registers 175° on an instant-read thermometer, about 45 minutes.
Transfer the chicken to a plate and let it cool until you can handle it easily.
Pour all the fat from the pan into a small bowl; do not rinse out the pan; the cooked juices will be part of the sauce.
Meanwhile, put the broth in a medium saucepan and bring it to a boil.
Add the carrots and cook until just barely tender, about 3 minutes; scoop them out with a slotted spoon and set aside.
Add the asparagus to the boiling broth and cook for about 30 seconds; scoop out and add to the carrots; set aside the vegetables and the broth.
When the chicken has cooled, pull off the skin, cut off the meat, and cut into small pieces; add to the vegetables.
Put the roasting pan over medium heat and add back about 2 tablespoons of the chicken fat.
Add the flour and whisk to make a smooth paste.
Cook for about 1 minute, whisking constantly, then slowly whisk in the broth to create a smooth sauce.
Cook the sauce, whisking frequently, until it is reduced to about 3 cups, is very smooth, and has the consistency of thick gravy, 10-15 minutes.
Remove from the heat, add the herbs, lemon zest, lemon juice, and a generous amount of pepper; taste for salt and add if needed.
Preheat oven again, at 400°.
On a lightly floured counter, roll the pastry just a bit larger than the size and shape of the top of your baking dish (at least 1 1/2 quart capacity); you can also make this using individual oven-proof bowls or ramekins.
Slide the pastry onto a baking sheet and bake until puffed and light brown, about 15 minutes.
Meanwhile, fold the cooked vegetables, the peas, and chicken into the sauce, and simmer gently to heat everything through.
When the pastry is ready, scrape the filling into the baking dish, transfer the pastry top over the filling.
Brush the top with a little heavy cream and sprinkle with grated Parmesan.
Return to potpie to the oven and continue cooking until the filling is hot and the pastry is deep golden brown, another 3-4 minutes.
**You can prepare the chicken, vegetables, and sauce one day ahead; refrigerate the sauce separately.