Preheat oven to 425°. In a large bowl, combine egg whites, 3 lbs. salt, 1/3 cup thyme, 1/4 cup lavender, peppercorns, and 1/2 cup water, mixing with your hands until salt is evenly coated and feels like wet sand.
Spoon enough of the salt mixture into a 9- by 13-in. baking pan to make a 1/4-in. layer in pan. Arrange beets on top without letting them touch. Mound remaining salt over each beet, pressing with hands to cover completely and make a crust.
Bake beets until very tender when pierced through salt, about 1 1/4 hours.
Poke tip of a table knife into salt crust about 1 in. from a beet and tap with mallet to crack crust. Move and tap knife in 1/2-in. increments in a circle around beet. Break beet free from crust and lift out. Repeat with remaining beets. Let beets stand until cool enough to handle.
In a medium bowl, whisk together olive oil, vinegar, shallot, pepper, and remaining salt, thyme, and lavender. Spoon 1 tbsp. dressing into a second bowl and 1/2 tbsp. dressing into a third bowl.
Brush remaining salt off beets with a pastry brush. Peel beets and cut into thin wedges. Add golden beets and avocado to largest amount of dressing. Add red beets to bowl with 1 tbsp. dressing, and mâche to bowl with 1/2 tbsp. dressing. Gently mix each to coat. Divide beets, avocado, and mâche among 6 salad plates.
Note: Nutritional analysis is per serving.
*Sodium content could not be accurately calculated.