Using a sharp knife, cut the top and bottom from the pomelo, slicing deep enough to reveal the fruit. Standing the pomelo on a cut end, cut off the peel, including the white pith. Working over a bowl, cut the segments free from the membranes on both sides. Once all of the segments are free, squeeze 1 Tbs. juice into the bowl. Cut the segments into bite-size pieces and return them to the bowl.
Put the flour in a wide, shallow bowl. In another wide, shallow bowl, whisk the egg with 2 Tbs. water.
Pat the chicken dry and generously season with salt and pepper.
In a heavy-duty 12-inch skillet, heat 2 Tbs. of the oil over medium heat until shimmering hot.
One at a time, quickly dredge 2 of the cutlets in the flour, then in the egg, letting the excess drip off, and transfer to the skillet.
Fry the cutlets, flipping once, until golden and just cooked through, 3 to 4 minutes per side. Transfer to a plate and repeat with the remaining oil and cutlets.
Add the wine to the skillet and boil until reduced by half, scraping the bottom of the skillet with a wooden spatula to loosen any browned bits, about 2 minutes. Add the chicken broth and boil until reduced by half, 2 to 3 minutes. Lower the heat to low. Add the butter and the pomelo pieces and juice and cook, stirring and shaking the pan, until the butter is incorporated, about 1 minute. Season to taste with salt and pepper.
Return the cutlets to the skillet to warm through, about 2 minutes. Serve the cutlets with the sauce spooned over them and sprinkled with the parsley.