Heat the millet in a small dry saucepan, with a tight-fitting lid, over medium-high heat, cook, shaking occasionally, until the millet begins to "pop". Continue toasting the millet until the popping subsides, about 2 minutes, until it smells like freshly popped popcorn. Set aside to cool slightly.
Add the water and the 1/4 teaspoon of the salt to the saucepan and bring to a boil over high heat. Wrap the pan's lid tightly with a small kitchen towel and cover the saucepan. (Make sure the towel's edges are folded up well away from the heat.) Lower the heat to low and simmer covered, for 20 minutes. Remove from the heat (don't uncover) and set aside for 10 minutes. Fluff the millet with a fork and transfer to a large bowl.
Using a box grater, grate the carrot and radish into the bowl of millet. Add the ginger, garlic, lime juice, mirin, and chili paste and stir to combine.
Meanwhile, bring a medium pot of water to a boil and salt it generously. Add the edamame, cover, and return to a boil; cook until soft, about 5 minutes. Drain and immediately add to the bowl of millet mixture (to lightly warm the vegetables). Stir to combine and set aside to cool.
When cool, season the edamame mixture with the remaining 2 teaspoons salt. Transfer to a food processor and puree into a paste. Return the edamame to the bowl and stir in the panko and egg whites until incorporated.
Using your hands, form the edamame mixture into 8 patties about 3 inches in diameter. Place on a parchment paper-lined baking sheet and refrigerate until set.
Place a rack about 4 inches from the broiler element and preheat. Place the patties on a foil-lined baking sheet or broiler pan. Brush both sides of the patties lightly with oil and sprinkle with salt. Broil until the tops are lightly browned, about 3 minutes. Remove from the oven and flip the patties with a spatula. Broil until lightly browned and hot, about 3 minutes more. Serve on grilled naan bread with sprouts, pickled ginger, and wasabi. (The burgers can also be grilled or sauteed.)
Mirin is a Japanese sweet rice wine available in Asian and specialty food stores.