Prepare the goat cheese: In a shallow dish, mix the bread crumbs, thyme (if using), salt and pepper. Crack the egg into a separate bowl and beat with a fork. Using unflavored dental floss or a sharp knife, cut the goat cheese into ½-inch-thick rounds.

Dip each round in the beaten egg, then dredge in the crumbs until completely covered, and transfer to a plate. Refrigerate until very firm, at least 5 minutes.

Meanwhile, make the salad: In a large bowl, whisk the vinegar, mustard and shallot with a pinch each of salt and pepper. While whisking, add the oil in a slow, thin stream and mix until emulsified. Add the greens and gently toss to coat. Season with more salt and pepper as needed. Divide among serving plates.

Fry the goat cheese: Heat a thin layer of oil in a large nonstick or well-seasoned cast-iron skillet over medium. Add the breaded goat cheese rounds and cook, turning once, until dark gold, 1 to 2 minutes per side.

Transfer to paper towels to drain, then set over the salad. Serve immediately.