Set out garlic herb butter to fully soften, if it isn't already. Putting your bread in the freezer while you prepare your other ingredients may help to get clean cuts.
Slice cucumbers and carrots into thin rounds (using a mandoline) or long strips (using a vegetable peeler).
Fill a medium bowl with ice and water. Bring a medium pot of water to a boil. Add carrot slices and blanch 30 seconds to 1 minute. Remove carrots and plunge into the ice bath to stop them from cooking further.
Remove carrots from ice bath once they're chilled, and pat them dry. Marinate the carrots and cucumbers in white wine vinegar, salt, and pepper for about 5 minutes. Drain, rinse, and thoroughly pat dry.
Spread the bread all the way to the edges with the fully-softened garlic herb butter. If making rectangular finger sandwiches with layered rounds, trim the crusts and cut bread into three rectangles (bottom).
Layer bread diagonally with alternating strips of carrots and cucumbers, or arrange alternating veggie rounds on the rectangles.
Trim the crusts off the veggie-striped bread.
If making striped sandwiches, first trim it into a square (not pictured). Then cut in half or quarters from corner to corner.