Preheat the oven to 150 degrees F.

To make the vegetable flakes, set a mandoline over a large bowl. Slice 1 fennel bulb paper-thin, then slice 1 onion, 1/2 of the celery root, and 1 of the apples to the same thickness. Switch to a vegetable peeler, and working over the same bowl, peel 1 carrot into thin ribbons. Mix the vegetables and spread them evenly on a baking sheet. Place the sheet in the oven to dry the vegetables until completely brittle, about 2 1/2 hours. (If you have an oven with a convection function, this will shorten the drying time to 1 hour.)

Transfer the vegetables to the work bowl of a food processor fitted with the metal blade and pulse until reduced to flakes. Store, covered, at room temperature. This can be done 1 day ahead.

Preheat the oven to 400 degrees F.

To prepare the birds, halve the remaining fennel, onion, carrot, and celery root, then cut them again into several small wedges. Place them in a large baking dish along with the potatoes and garlic. Generously season with salt and pepper. Drizzle with 3 tablespoons of the olive oil and mix to coat.

Season the guinea fowl with salt and pepper inside and out. Stuff the cavities with whole sprigs of thyme. Rub them all over with the remaining 2 tablespoons of olive oil. Place the birds on the bed of vegetables. Roast, uncovered, for 45 minutes, basting occasionally with pan juices.

Cut the remaining apple into several wedges and add it to the roasting pan. Continue roasting and basting until the skin on the birds turns golden brown, another 20 minutes. Remove the birds from the oven, sprinkle them generously with dried vegetable flakes, then continue to roast until the juices run clear, another 10 minutes. Transfer the birds and roasted vegetables to a serving platter. Cover loosely with foil and let rest while preparing the sauce.

To make the pan sauce, pour the pan juices into a small saucepan. Skim and discard as much fat off the top as possible. Over medium-low heat, deglaze the roasting pan with the cider and simmer for 5 minutes, scraping the bottom of the roasting pan to loosen the browned bits. Add the hot cider from the roasting pan to the pan juices, and over high heat, reduce the liquid by half. Stir in the butter, season with salt and pepper, and pour it into a sauce boat. Sprinkle the birds and the vegetables with persillade. Carve the guinea fowl at the table and serve with the cider sauce.