Carefully slice the goat cheese into 12 small rounds about one-third-inch thick.
Combine the egg, 1 tablespoon peanut oil and 1 tablespoon cold water in a small bowl and mix with a whisk. Spread the bread crumbs over a plate and season with salt and pepper. Dunk the cheese rounds, one at a time, into the egg mixture, letting the excess drip back into the bowl, then roll in the bread crumbs to fully coat on both sides. Refrigerate for about 15 minutes.
Heat the butter and the remaining 3 tablespoons peanut oil in a skillet over medium-high heat. Carefully place the breaded cheese rounds in the pan and fry until light brown, about 2 to 3 minutes on each side. Drain on paper towels.
For the vinaigrette: Place 1 tablespoon vinegar in a bowl, season with salt and pepper, and slowly beat in the olive oil and hazelnut oil.
To serve, combine the salad greens, vinaigrette and chopped walnuts in a bowl. Place a small mound of salad greens in the center of 4 plates, surround each with 3 cheese rounds, drizzle the outside of the plate with balsamic vinegar, and garnish with the chives.
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