Prepare a one-quarter sheet of silicone paper by poking it with a needle, making as many holes as you can. The silicone paper needs to be perforated because Methocel requires coming into contact with heat on its entire surface in order to gel. If you don’t perforate the paper you will end up with a crisp surface but a soft base.
Turn a dehydrator on to 65°C/150°F.
Combine the xanthan gum, Methocel F50, and sugar in a coffee grinder and grind to obtain a fine homogenous mix.
Using a handheld immersion blender, shear the dry mix into the carrot juice in the bowl of an electric mixer, pouring the dry ingredients slowly to avoid clumping. You should have a thick, viscous mass.
Place the bowl on an electric mixer fitted with the whip attachment. Whip on high speed until the liquid increases in volume eightfold and looks like shaving cream.
Spread the foam into an even layer on the perforated silicone paper. It should be about 2.5cm/1in thick.
Place in the dehydrator and let it dry for at least 2 hours.
Once the foam has gelled, it will remain soft if it is warm, but when it cools it will be crisp. Reserve this foam warm in a dehydrator until ready to assemble the entremets. Discard after 1 week.