SLICE each goat cheese log into 6 1/2-in. rounds. You should have 12 cheese rounds. Whisk eggs in a shallow bowl. Pour panko into another shallow bowl. Dip cheese rounds in egg, then in panko until well coated. Place on a large plate. Cover with plastic wrap and freeze until firm, 2 hours or overnight.
HEAT a small non-stick frying pan over medium-high. Add 2 tbsp oil, then 3 frozen cheese rounds. Cook until golden, about 2 min per side. Transfer to a paper-towel-lined plate. Repeat with remaining cheese.
WHISK remaining 2 tbsp oil with vinegars and salt in a large bowl. Season with fresh pepper. Add greens and toss until coated. Divide among 6 plates. Top each with 2 cheese rounds.