Make the ice cream. Put all of the ingredients in a blender and blitz until totally smooth. Make sure there are no clumps of milk powder hiding in the mix.
Next, pour the mixture into the frozen base of an ice cream maker and churn according to the manufacturers directions. Scoop the ice cream into a storage container, cover tightly and freeze until more solid, at least 6 hours.
Make the cookies. Preheat an oven to 400 degrees F and line two baking sheets with parchment. Set aside.
Whisk the flour, xanthan gum (if using), cream of tartar, baking soda and salt in a medium bowl.
Put the sugar and lavender in the bowl of a food processor fitted with a steel blade attachment. Pulse until the lavender is finely ground. Pour the sugar through a sieve into a large bowl and discard any lavender bits that are left behind. Add the melted butter, eggs, lemon zest, vanilla extract and lemon extract to the sugar and beat together until smooth.
Stir in the flour mixture and beat until it is evenly incorporated and dough is smooth. If the dough feels too sticky to handle at this point, you can put it in the freezer for about 15 minutes so it firms up. Using a cookie scoop or a spoon, scoop the dough into balls that are about the size of a golf ball. Roll the dough balls in the sugar to coat and place on the cookie sheet with a few inches between each cookie. If you want puffy cookies, leave as they are. If you prefer flatter cookies (pictured above) smash the dough until it’s about 1/4 of an inch thick using the bottom of a glass. Sprinkle with yellow sprinkles if you’re feeling WILD!
Bake for about 10 minutes or until the cookies are golden brown around the edges.
Once the cookies are cool and the ice cream has chilled, place a large scoop of ice cream on the upside down side of one snickerdoodle, top with another cookie and gently press to flatten. Freeze the cookies on a baking tray until solid, then tightly wrap with a layer of plastic, and a layer of foil and store in the freezer until ready to eat!