Combine vinegar, walnut oil, mustard, shallot, and 2 tablespoons olive oil; set vinaigrette aside. Combine egg and water in a shallow bowl. Combine walnuts and breadcrumbs in a shallow dish. Cut goat cheese into 12, 1/4-inch rounds. Dip in egg mixture, and dredge in walnut mixture. Freeze 30 minutes.
Sauté goat cheese rounds, in batches, in remaining 1 tablespoon olive oil in a nonstick skillet over medium-high heat 1 minute on each side or until golden brown.
Arrange tomatoes and goat cheese on top of arugula. Drizzle with vinaigrette.