How to Make It
Place saffron threads in a small skillet; cook over medium, undisturbed, until toasted, about 1 minute. Remove from heat; crumble into a small bowl. Set aside.
Fold an 8-inch-square piece of cheesecloth into quarters to make a 4-layer, 4-inch square. Using a Microplane grater, grate ginger over center of cheesecloth square. Gather edges of cheesecloth together, and squeeze ginger over a small bowl to equal 2 tablespoons ginger juice. Discard solids. Repeat process with turmeric root.
Place egg yolks in a food processor. With food processor running, add lemon juice, crumbled saffron, ginger juice, turmeric juice, and 1/2 teaspoon salt; slowly stream in neutral oil until mixture is smooth and combined, 8 to 10 minutes. Transfer aioli to a small bowl.
Pour canola oil into a Dutch oven to a depth of 3 inches; heat over high to 350°F.
Meanwhile, stir together togarashi and smoked paprika in a small bowl; set aside.
Stir together sour orange juice and 21/2 teaspoons salt in a large bowl. Add mushroom pieces, gently pressing to submerge. Let soak 2 to 3 minutes. Drain well, and discard orange juice mixture.
Place cornstarch in a bowl. Place 2 of the mushroom pieces in cornstarch; toss to coat. Using a spider, transfer mushrooms to hot oil in Dutch oven. Fry until golden brown, about 5 minutes, turning once halfway through cook time. Transfer fried mushrooms to a plate lined with paper towels. Repeat process with remaining 2 mushroom pieces. Sprinkle mushrooms evenly with remaining 1/2 teaspoon salt.
To serve, spread 2 tablespoons aioli on each of 4 plates; spoon remaining aioli into a small bowl. Top plates evenly with mushrooms; sprinkle with chives and togarashi-paprika mixture. Serve with aioli in bowl.
If sour oranges are unavailable, a mixture of 1 cup fresh orange juice and 1/2 cup fresh lime juice may be substituted.