Spread out Brussels sprouts, lemon, capers, and garlic in a single layer on a paper towel–lined baking sheet and set aside for 30 minutes at room temperature. (This will ensure the ingredients are dry before frying.)
Fill a large, wide pot with 2 inches of vegetable oil. Bring oil to 400°F over medium heat.
Working in batches, add Brussels sprouts to oil and stir slowly to avoid splashing. Cook until sprouts float and are golden brown, about 3 minutes, then remove with a slotted spoon and place on a paper towel–lined baking sheet. Season with salt.
Bring oil back to 400°F before repeating process with remaining sprouts.
Return oil to 400°F and add lemon, capers, and garlic. Stir and fry for 1 minute. Remove with slotted spoon and place on paper towel–lined baking sheet. Season with salt.
Transfer fried ingredients to a large bowl and toss with lemon juice, red pepper flakes, and burrata. Taste and adjust seasoning as necessary with salt and red pepper flakes. Serve immediately.