To prepare salad, place potatoes in a medium saucepan; cover with water. Bring to a boil. Reduce heat; simmer 10 minutes or until tender. Drain; rinse with cold water. Drain. Place potatoes in a large bowl.
Bring 4 cups water to a boil in a medium saucepan; add cauliflower and broccoli. Cook 1 minute or until crisp-tender. Drain; rinse with cold water. Drain. Add cauliflower mixture to potatoes.
Using a vegetable peeler, scrape carrots from tip to stem to make long ribbons. Add carrot to potato mixture, tossing gently to combine.
Leave root and 1-inch stem on beets; scrub with a brush. Place in a medium saucepan; cover with water. Bring to a boil. Cover, reduce heat, and simmer 25 minutes or until tender. Drain; rinse with cold water. Drain; cool. Trim off beet roots; rub off skins. Cut beets into 1/4-inch-thick slices. Arrange lettuce and cabbage leaves alternately on a large platter.
To prepare vinaigrette, combine thyme and next 5 ingredients (thyme through 1/8 teaspoon black pepper), stirring with a whisk. Add oil, stirring well to combine; drizzle vinaigrette over lettuce. Spoon potato mixture over lettuce; top with beets.
To prepare dipping sauce, place chiles in a spice or coffee grinder; process until finely ground. Place 1/8 to 1/4 teaspoon ground chile in a medium bowl (reserve remaining ground chile for another use). Add cocktail sauce and remaining ingredients, stirring well to combine. Serve with vegetables.