Slice the goat cheese into four 2-oz. disks. Shape the disks into 2-1/2-inch cakes. Freeze in a single layer until firm, about 10 minutes.
Combine the panko, parsley, mustard, 1/2 tsp. salt, and 1/2 tsp. pepper in a small bowl. In another small bowl, beat the egg with 1 Tbs. water.
Dip the goat cheese cakes in the egg mixture, shaking off any excess, and then in the panko mixture. Repeat, coating each cake twice.
Heat the vegetable oil in a 12-inch skillet over medium heat until shimmering hot. Fry the cakes, flipping once, until golden-brown on both sides, 3 to 4 minutes total. Transfer to a paper-towel-lined plate.
Whisk the vinegar, maple syrup, 1/4 tsp. salt, and 1/4 tsp. pepper in a small bowl. Slowly whisk in the nut or olive oil until emulsified.
Toss the radicchio, arugula, and grapes in a large bowl with enough dressing to lightly coat. Divide the salad among 4 plates, top each with a goat cheese cake, sprinkle with the hazelnuts, and drizzle with more of the dressing.