Use a cheese grater to grate the ginger into small pieces. You should have about 2/3 cup at the end of this. Add it to a small pot with the filtered water. Bring to a low simmer and cook for about 20 minutes. Let the mixture cool on the stove.
Strain out the ginger pieces through a fine mesh strainer - you should have about 1 to 1 1/4 cups of ginger juice after simmering. Whisk in honey or maple syrup and cardamom. If you want to use this base for ice cubes, freeze it in an ice mold here.
Fill a glass with ice. Combine about 1/4-1/3 cup of the ginger ale base with about 1/2 cup of sparkling water or club soda. Squeeze 1/2 lime into it, if using, and stir together.
Notes
If you end up with less than 1 cup of concentrated ginger juice after simmering, you either need to turn down the heat (too high/boiling) or it has simmered too long. You can add a little more water if you need to!
Extra strong ginger ale - use this homemade ginger juice as a base instead of boiling.
Maple syrup can be substituted for honey.
Leftover ginger can be frozen into small cubes and used for cooking/ice cubes.
Lime is not an allowed food on a migraine elimination diet, so omit if you are in the elimination phase or sensitive to citrus.
Ginger ale ice cubes - Make the ginger ale base through step 2 and freeze in an ice cube mold for 6-12 hours (or overnight). Then add to sparkling water or your favorite sparkling beverage.
Tart cherry "sleepytime" ginger ale - Add 1 tablespoon of tart cherry juice to the mixture.