Preheat oven to 250°F. Prepare pig’s ears by removing any hair with a sharp knife or razor. Place the ears in a just-big-enough ovenproof saucepan and add enough duck fat to cover. Cut a circle of parchment paper and lay it over the ears, then put a saucer on top to weigh everything down. Cover the pan and cook in the oven until the ears are tender enough that when you give a firm pinch, your fingers touch, about 4 hours.
Prepare the Lemon Caper Dressing: Segment the lemons and set the segments aside. Squeeze the juice from the membranes into a separate bowl and add the rest of the ingredients, stirring well. Add the lemon segments and gently toss to coat without breaking them. Use immediately or chill in the refrigerator for up to 1 hour.
Remove the ears from the oven and let them cool in the fat. Once cool, remove from the pan and wipe them off, then pat dry. Strain and reserve the duck fat for another use.
Prepare the salad by separating the leaves of your endive, Treviso or radicchio. Put them in a large bowl, cover with plastic wrap and chill.
Heat the peanut oil in a deep fryer to 350°F. Be sure to have a lid ready to protect against spatter. Carefully and gently add 1 ear (or 2 if your fryer is large) and immediately close the lid. Fry until golden brown, about 8 minutes. Transfer to paper towels to drain and season immediately with salt. Keep the ear(s) in a warm place while you fry the rest.
Remove the chilled greens from the refrigerator and spoon 5 tablespoons of the chilled dressing evenly over the leaves. Add the arugula and toss gently with your hands, taking care to coat the leaves without breaking up the lemon segments in the dressing.
Plate the greens on a platter and top with the warm ears, adding a bit more dressing if you like.