Fill a large bowl with ice cubes. Mix flour and arrowroot powder together a smaller bowl; set over the ice cubes. Pour ice water slowly into the flour mixture, mixing until the batter has a medium-thick texture.
Heat oil in a deep-fryer or large saucepan over medium-high heat. Dip each piece of cauliflower into the batter and cook in batches in the hot oil until golden brown, 1 to 2 minutes. Place on paper towels to absorb excess oil. Repeat with broccoli, red bell pepper, and baby carrots.