For the Goat Cheese. Process Melba toasts in food processor to fine, even crumbs, about 1 1/2 minutes; transfer to medium bowl and stir in pepper. In separate medium bowl, whisk eggs and mustard together. In third medium bowl, combine thyme and chives.
Using cheese wire or dental floss, slice cheese into 12 equal-size pieces. Roll each piece of cheese into ball, then roll in herbs to coat lightly. Working with 6 pieces at a time, roll in egg mixture, then coat with crumbs, pressing to help coating adhere. Flatten each ball into 1-inch-thick disk and lay on large plate. Freeze cheese until firm, at least 30 minutes. (Breaded goat cheese can be kept in freezer for up to 1 week.)
For the Salad: Soak dried figs in 1/2 cup hot water in bowl until plump, about 10 minutes; drain. Combine vinegar, mustard, shallot, ¼ teaspoon salt, and sugar in large bowl. Whisking constantly, drizzle in oil. Season with pepper to taste.
Adjust oven rack to top position and heat oven to 475 degrees. Brush tops and sides of cheese evenly with oil. Bake until crumbs are golden brown and cheese is slightly soft, 7 to 12 minutes. Using thin metal spatula, transfer cheese to paper towel–lined plate and let cool for 3 minutes.
Combine greens and dressing in large bowl, toss gently to coat, and divide among salad plates. Sprinkle with figs, apples, and walnuts. Arrange warm goat cheese on top of salad and serve.