How to Make It
Prepare the mushrooms: Preheat oven to 400°F. Melt ½ cup of the duck fat in a small saucepan over high until liquid, about 2 minutes.
Combine mushrooms, thyme, salt, pepper, and garlic in a large bowl; toss to coat. Arrange mushroom mixture in a single layer on a baking sheet lined with parchment paper. Roast in preheated oven until deeply caramelized, 40 to 45 minutes, stirring occasionally. Squeeze garlic cloves into a food processor; discard papery skins. Add mushroom mixture, vinegar, and remaining 2 Tbsp. duck fat. Pulse until very finely chopped. Set aside.
Prepare the burgers: Place all burger ingredients in a stand mixer fitted with a paddle attachment. Beat on low until miso is incorporated. Form mixture into 4 (¾-inch-thick) patties.
Prepare the togarashi mayonnaise: Stir together all mayo ingredients in a bowl.
Preheat grill to medium-high (400°F to 450°F). Place burgers on oiled grates; grill, uncovered, turning occasionally, until desired degree of doneness, 8 to 10 minutes for medium.
Divide mayo mixture among top and bottom halves of rolls. Place burgers on bottom rolls. Top with mushroom mixture, watercress, and top halves of rolls.