Peel the lotus root and slice it about 2 mm thick, halving the rounds into half-moons if the root is wide. Drop the slices into a bowl of cold water and leave for a few minutes, then drain well. This stops them browning and washes off the surface starch.
Cut the carrot into thin matchsticks, about the same size as the lotus root slices so they cook in the same time.
Heat the oil in a frying pan over medium heat with the dried chili rings in it from the start.
Add the drained lotus root and stir-fry for 2 to 3 minutes, until the edges turn translucent and pick up a little color.
Add the carrot and stir-fry another 1 to 2 minutes, until it softens but still has bite.
Add the soy sauce, mirin, and sugar all at once. Keep the pan moving until the liquid reduces and coats the vegetables in a glossy glaze, 1 to 2 minutes. The pan should be almost dry.
Take off the heat, stir through toasted sesame seeds, and serve warm, at room temperature, or cold.
