How to Make It
Prepare the fennel-celery slaw: Combine all slaw ingredients in a bowl; toss to coat. Chill until ready to use.
Prepare the burgers: Gently fold together all burger ingredients in a bowl until combined. Form into 4 (¾-inch-thick) patties.
Preheat grill to high (450°F to 500°F). Place burgers on oiled grates. Grill, uncovered, to desired degree of doneness, 3 to 4 minutes per side for medium.
Prepare the tempura lemons: Heat oil to 350°F in a heavy saucepan. Whisk together flour, vodka, club soda, and egg yolk in a bowl. (It's OK if it's slightly lumpy; don't overwork it.) Dip lemon slices in batter. Carefully place lemons in hot oil, and fry until they float and are crispy, 1 to 2 minutes. Drain on paper towels, and sprinkle with salt.
Place 1 burger on each of the bottom buns. Top each with 2 lemon slices, 1/3 cup slaw, and top bun.