Melt butter in a small saucepan over low heat (fat will rise to the top, and milk solids will sink to the bottom). Skim off white froth that appears on top. Strain off the clear, yellow butter, discarding milk solids. Set aside.
Drain and pat grated potato and celery root dry, pressing between paper towels. Combine grated vegetables, corn, and next 4 ingredients in a large bowl. Stir in flour and egg. Shape vegetable mixture into 24 (2-inch) minicakes.
Heat about 2 tablespoons clarified butter in a large nonstick skillet over medium heat. Cook 8 cakes, 7 to 8 minutes on each side or until golden. Repeat procedure with remaining butter and cakes. Place a 1/2-ounce piece of smoked trout on each cake. Top with Horseradish Crème Fraîche and Golden Beet Relish. Serve warm.