In a large bowl, add the warm water. Sprinkle on the yeast and sugar, then cover the bowl with a kitchen towel, let it rest and become bubbly and creamy, about 5 minutes.
Meanwhile, set a fine meshed sieve or sifter over a medium bowl. Add to the sieve the almond flour, psyllium husk, flaxseed, baking powder, xanthan gum, and salt and tap it until the contents sift through, pushing through any remaining bits. Set aside.
In a small bowl, add the egg, egg whites, olive oil, and vinegar; whisk to combine. Set aside 1 tablespoon of the mixture in a small bowl to brush bagels with later. Add the remaining egg mixture to the proofed yeast, whisking until light and frothy. Working in two batches, add the flour mixture to the yeast-egg mixture, mixing quickly to incorporate and create a thick, sticky dough.
Line a rimmed baking sheet with a silicone baking mat or parchment paper. With wet hands, divide the dough into 6 equal rounds, about 2½ ounces each. Smooth the rounds as much as possible with your hands, then make a hole in the center with your index finger, gently stretching out the dough and forming into a bagel shape, about 3-inches in diameter.
Use about 1 teaspoon oil to grease one side of a piece of plastic wrap that’s big enough to cover the baking sheet pan. Cover the pan with the plastic wrap, oiled-side down, and place it into a warm draft-free space for 60 minutes to allow the flavors to develop.
Preheat the oven to 350°F (180°C). Move an oven rack to the bottom third of the oven.
Brush the bagels with the reserved egg mixture and sprinkle them with the everything seasoning.
Bake until deep golden, 22 to 25 minutes, checking on the bagels after about 15 minutes and tenting with aluminum foil if the top or toppings are getting too brown. You don’t want the sesame seeds getting any darker than light golden brown.
Allow the bagels to cool on the pan for 1 hour, then transfer to a wire rack to let cool completely, about 2 hours.