In a medium bowl, whisk together by hand the flours, xanthan gum, garlic powder, salt, baking powder, and baking soda.
In a large bowl, beat the egg whites at high speed until stiff peaks form. (Use whisk attachment on stand mixer.)
In another bowl (or the same bowl if you've carefully transferred the beaten egg whites to a separate clean bowl), beat the lard (with the flat edge beater attachment).
Add the three egg yolks and continue to beat.
Add about ¼ cup of the dry flour mixture to the lard-egg mixture. Beat until smooth and creamy.
Add remaining dry mixture and blend at medium-low speed until combined. Stop to scrape down the sides of the bowl if necessary.
Add water and vinegar and beat until smooth.
Add about ⅓ of the beaten egg whites to the lard-flour mixture and beat at low speed until just blended.
By hand, gently fold in the remaining egg whites.
Cut a 1-inch diameter opening across a bottom corner of an icing bag or a plastic freezer bag. Gently transfer the batter into the bag. Handle the batter carefully at this point so as not to break air bubbles unnecessarily.
Using the bag of batter, pipe "sticks" across the width of the prepared pans.
Bake in preheated 325 F oven for 1 hour. Then, if necessary, turn up the heat and bake for up to 15 minutes longer until the bread sticks are nicely browned. (Cooking time will vary depending on the thickness of your bread sticks, so watch them closely near the end.)
Bread sticks will keep in the refrigerator for several days or in the freezer for a few months. Simply freshen them up by popping them into a 350F oven for 5 to 10 minutes.