In a small bowl, mix together the mayonnaise, lime juice, and five-spice powder. Refrigerate the aioli until ready to use.
For the relish:
In a saute pan over medium-high heat, toast the coriander seeds until fragrant. Drizzle in the vegetable oil, add the red onion and cook, stirring frequently, until translucent. Stir in the chile and white wine vinegar, and then sprinkle over the sugar, stirring until it dissolves.
Add the cucumber to the pan, and cook for 5 to 7 minutes over medium heat, and reduce slightly, achieving a light glaze consistency. Season the relish with salt, and let the mixture cool.
Just before serving the relish, stir in the mint, and season with salt, to taste.
For the burgers:
Put the ground lamb, shallot, fresh ginger, garlic, egg yolk, and five-spice powder in a bowl, and mix well. Season the mixture with salt, and black pepper. Divide and shape the lamb mixture into 4 (8-ounce) burger patties. Place the patties in the refrigerator for 1 hour, to allow the flavors to infuse.
Preheat a grill to medium-high and lightly oil the grate. Grill the lamb patties until they are seared on both sides, and cooked through. Remove the patties from the grill, and set aside. Place the buns on the grill, cut-side down, and toast until golden.
Put the lamb patties on the toasted buns, slathered with the Five-Spice Aioli, Pickled Cucumber Relish, and sliced tomato.