The brussels sprouts should be thoroughly dry before frying. Any water clinging to the sprouts will cause the oil to splatter. The oil takes time to come up to temperature, so get that heating on the stove top before prepping the other ingredients.
In a heavy-bottomed pot, heat 2 inches of oil over medium-high. When oil reaches 360 degrees on a thermometer, carefully add 1/3 of the brussels sprouts. Cook, turning occasionally, until crisp and golden brown, 2 to 3 minutes. Transfer to a paper towel-lined tray with a slotted spoon and sprinkle with salt. Fry remaining sprouts in batches, returning oil to 360 degrees between each batch. Add shallots and chiles to oil and fry until slightly wrinkled and just beginning to color, 1 to 2 minutes. Transfer to tray with sprouts.
Whisk lime juice and maple syrup in a large bowl. Add brussels sprouts, shallots, and chiles, tossing to coat evenly in lime mixture. Season with more salt to taste. Serve immediately.