Whisk together egg white and water. Dip cheese rounds in egg, letting excess drip off, then dredge in bread crumbs, pressing lightly to adhere. Chill for 15 minutes.
Whisk together vinegar, salt, mustard, sugar, and a pinch of pepper in a small salad bowl. Add 1 tablespoon oil in a slow stream, whisking until emulsified.
Heat remaining teaspoon oil in a small nonstick skillet over moderately high heat until hot but not smoking, then saute cheese until golden, about 30 seconds per side. Remove skillet from heat.
Add greens to dressing and toss gently to coat. Serve salad topped with warm cheese.
Cooks' note: The easiest way to cut goat cheese is with a piece of unflavored dental floss.