Preparing to bake: Heat the oven to 375 degrees F. Lay down greased parchment paper into two 9-inch cake pans.
Combining the dry ingredients: Whisk together the flour mix, psyllium, baking powder, and salt. Set aside.
Mixing the wet ingredients: Stir together the milk and vanilla.
Creaming the butter and sugar: Put the butter into the bowl of stand mixer. Whirl the butter until it has spread out in the bowl. Add the sugar. Mix the butter and sugar together until the sugar has entirely disappeared into the butter and has come creamy.
Finishing the batter: With the stand mixer running on low, add 1/3 of the dry ingredients. When there is no flour visible, add 1/2 the milk. Repeat, ending with the last of the dry ingredients. Carefully, scrape the cake batter into a large bowl.
Beating the egg whites: Clean out the bowl of the stand mixer and dry it thoroughly. Add the egg whites. Turn on the stand mixer, with the whisk attachment on. Add the cream of tartar to the egg whites. Let the stand mixer run until the egg whites turn white and start to thicken. Pour in the sugar and continue whisking the egg whites until they have reached soft peaks.
Folding in the egg whites: Pour 1/3 of the egg whites into the cake batter. Slowly, with sure strokes, use a rubber spatula to fold the egg whites into the cake batter. When there is no visible sign of egg whites left, pour another 1/3 of the egg whites. Continue doing this until the cake batter is fluffy and increased in volume. Stir in the last and stop stirring.
Baking the cake: Pour the cake batter equally into the 2 cake pans. Slide them into the oven and bake until the top of the cakes has a little athletic jiggle, and a toothpick inserted into the middle comes out clean, 30 to 40 minutes.
Allow the cakes to cool in the pan. Run a butter knife around the edges of the cake pans to loosen the cake. Carefully, turn the cakes onto cooling racks.
Pour all the flours into a large container. (Restaurant supply stores sell large plastic containers that fit this purpose well. You could also use a large glass jar.) Shake and shake and shake harder until all the flours have become one color.