For the fennel juice: Place the ascorbic acid and the salt in a one-sixth pan. This will be the pan into which the fennel juice will be pouring as it comes out of the juicer.
Juice the fennel. Stir the juice with a spoon or whisk so that the ascorbic acid and the salt dissolve.
Strain the liquid through a fine-mesh sieve, then pass it through a fine-mesh bag (commercially known as a Superbag; see Resources). Stir in the sugar until it has dissolved.
For the meringue: Line a half hotel pan with a sheet of acetate. Pour the fennel juice into a small electric mixer bowl.
Pour the egg white powder over the fennel juice and mix on low speed.
Meanwhile, make a syrup with the sugar and the water by combining them in a small sauce pot and bringing them to a boil. Turn off the heat and make sure that all of the sugar has dissolved. Add the gelatin and stir until it has dissolved. Let it cool to room temperature.
Add this syrup to the juice and egg white powder mixture when it has reached half the maximum volume (it will have tripled in size). Whip on high speed until the mixture reaches full volume (it will have increased in volume six times from its original volume), and then pour in the absinthe. Mix for 10 more seconds, and then pour the foam into the prepared hotel pan and place in the freezer to harden.
Once hardened, reserve frozen for service or wrap tightly and reserve frozen. Discard after 1 week.