To make the pork marinade, combine the onion, garlic, orange juice, lime juice, and salt in a resealable gallon-sized plastic bag. Add the pork, turn to coat in the marinade, and refrigerate 1 hour or up to overnight.
Meanwhile make the chutney. In a saucepan over medium heat, combine the vinegar, rum, raisins, brown sugar, red onion, bell pepper, ginger, garlic, and allspice. Bring to a boil, stirring occasionally. Reduce the heat to medium-low and simmer until the onions and peppers are soft, 8 to 10 minutes. Add the mangoes and the apple and continue to simmer, stirring occasionally, until the fruit begins to break down and the mixture is thick and jammy, about 15 to 20 minutes. Turn off the heat and set aside to cool. Cover and refrigerate for up to 1 week.
Spread the flour on a plate and season with salt and pepper. Put the eggs in a shallow bowl and lightly beat. Spread the bread crumbs out on another plate and season with salt and pepper. Remove the pork from the marinade and pat dry. Dredge both sides of the cutlets in the flour, and then dip into the egg, letting the excess drip back into the bowl. Press both sides of the cutlets in the bread crumbs so they stick. Set the pork cutlets on a platter and set aside for 30 minutes to allow the breading to set.
Heat the oil in a large skillet over medium heat. Fry the cutlets until they are golden brown, 3 to 4 minutes on each side. Transfer to a paper towel-lined plate to drain and serve with the Mango-Rum Chutney.