Making a flute is probably easier than a whistle if your tools are a carrot and a drill. Whistles need a flat aperture and a slanted sharp edge to blow the air across (ask me how I know!).
For a flute, all you need is a slightly larger top hole to blow across sideways, a bit like making a bottle sound. The trick with the mouth hole on a flute is to have the edges sharp, at least the outside one. Then the sound would travel down the bore. No idea whether a carrot would be resonant enough either. Atlernatively, you could take the plastic fipple off a whistle and install the bored-thru carrot.
I used to make flutes out of PVC pipe, and stick a cork in the end -- the distance between the interior end of the cork and the blow hole is crucial. To make a flute play, place the curve of the barrel -- with the mouth hole on top -- in the hollow below your lower lip, and then rotate the flute slightly back and forth until you hear a noise.
Have fun!
Blessings,
Barbara