For the lotus chips: Separate the lotus root segments by snapping them apart and wash well. Trim the stem ends and peel the root with a vegetable peeler. Using a mandoline, slice crosswise into 1/4-inch slices and soak in cold water to remove the starch and any remaining dirt. Rinse again, dry in a salad spinner, and blot dry on paper towels.
Heat the vegetable oil to 350°F in a deep-fryer or wide deep pot and line a sheet pan with paper towels. Fry until golden brown and crispy. Transfer to the sheet pan, drain briefly on the paper towels, and sprinkle with salt.
For the rice-cake buns: Combine the rice and water in a saucepan over medium-high heat. Bring the water to a boil, cover with a tight-fitting lid, reduce the heat to low and simmer until the water is absorbed, 15 to 20 minutes. Remove from the heat and leave covered for 15 minutes.
Pour the rice into a large bowl, fluffing it with a fork, and adding the remaining ingredients. Let cool.
Brush four half sheet pans with peanut oil. Divide the rice between two of the pans, pressing to 3/4-inch thickness. Using a lightly oiled 3-inch circular cutter, cut 12 rounds from each pan and arrange on the remaining sheet pans. Lay parchment paper over the top and refrigerate overnight, allowing the rice to dry and set.
For the glaze: Stir all of the ingredients together in a nonreactive container.
For the chicken burgers: With gloved hands, mix together all of the ingredients until well combined. Divide into twelve equal portions (about 5-ounces each) and shape into 4-inch patties.
To complete: Pour a generous film of peanut oil into a large sauté pan and heat over medium heat. Working in batches, fry the rice-cakes until golden brown on one side. Transfer to a sheet pan. (The rice-cakes can be re-crisped in a warm oven if needed.)
Wipe out the pan, add enough oil to coat, and set over medium-high heat until the oil is rippling. Working in batches, cook the chicken patties, turning once, until the internal temperature is 165°F, about 10 minutes.
Reserve 1/4 cup of the glaze for serving. Brush the remaining glaze on both sides of the patties. Return the patties to the pan just to set the glaze.
To serve: Set a rice cake crisped side down on each plate. Top with a chicken burger, drizzle with glaze, top with 2 tomato slices, arugula, and another rice-cake, crisped side up. Serve with the lotus chips on the side.