How to Make It
In a small nonreactive saucepan, bring the sugar, vinegar and lemon juice to a boil over high heat. Cook until golden and reduced to 1 1/2 tablespoons, about 3 minutes.
In another small saucepan, heat 1/2 teaspoon of the olive oil. Add the onion and carrot and cook over moderate heat, stirring, until softened, about 4 minutes. Add the sugar-vinegar syrup, fresh ginger, and chicken stock and boil over high heat until reduced to about 1/2 cup, about 15 minutes. Strain the sauce into a small bowl. Stir in the julienned pickled ginger and lime zest.
Heat the remaining 1 tablespoon olive oil in a large, heavy, nonreactive skillet. Season the pork on both sides with salt and pepper. Add the medallions to the skillet, in batches if necessary, and cook over high heat until browned and crisp, about 3 minutes per side for medium. Arrange the pork on 4 plates and keep warm.
Add the ginger sauce to the skillet and bring to a simmer over high heat, scraping up any browned bits. Spoon the sauce over the pork and garnish with the lime wedges. Unmold an Apple-Onion Flan onto each plate and serve at once.