1. Activate the Yeast: In a mixing cup, combine the warm heavy cream, water, and inulin. Sprinkle the active dry yeast over the top, give it a quick stir, and set it aside to bloom for about 5 minutes until it looks foamy.
2. Combine Dry Ingredients: While the yeast is activating, in the clean, dry bowl of a stand mixer (or a large mixing bowl if you're mixing by hand), add the vital wheat gluten, oat fiber, 1 cup of golden flaxseed, sweetener, baking powder, and salt.
3. Whisk Well: Whisk these dry ingredients together thoroughly until they are uniformly combined.
4. Add Yeast Mixture: Attach the paddle to your stand mixer. On a low speed, slowly pour in the activated yeast mixture.
5. Incorporate Eggs: Continue mixing on low, gradually pouring in the beaten eggs and egg yolk until well combined.
6. Add Butter Gradually: Add the softened butter, one teaspoon at a time, ensuring each addition is fully incorporated into the dough before adding the next.
7. Adjust and Knead: Check the dough's stickiness. If it's too sticky, gradually add more golden flaxseed, up to an additional 1/4 cup, until the dough is slightly tacky but not overly sticky. Switch to a dough hook attachment and knead the dough on medium-high speed for 10 minutes until it's smooth and elastic.
8. Shape Dough Logs: Transfer the dough onto a clean board or a Silpat. Divide it into 3 equal pieces, shaping each into a neat log. For accuracy, you can weigh each piece (around 10.3 ounces per log is a good guide).
9. Rest the Dough: Cover the dough logs with plastic wrap and let them rest for 15 minutes.
10. Form Ropes: Roll each dough piece from the inside out into a uniform 16-inch rope.
11. Braid and Pan: Stack the ends of the three ropes together and pinch them closed. Then, carefully braid the ropes, pinching the other ends to secure them. Place the braided dough into a 9-inch loaf pan that's been lined with parchment paper and lightly sprayed with olive oil.
12. Proof for Rise: To help the dough rise beautifully, cover the pan and place it in a dark, warm environment (like your microwave or an oven set to proofing mode). For extra humidity, place a bowl of boiling water alongside it.
13. Allow to Rise: Let the dough rise for 2 hours, or until it has more than doubled in size and looks wonderfully puffy.
14. Preheat Oven: Once risen, gently remove the covered dough from its warm spot and set it on your counter while you preheat your oven to 375°F (190°C).
15. Bake to Golden: Remove the plastic wrap and brush the top of the dough with the egg wash. Bake for 25-30 minutes, or until the brioche is golden brown and cooked through. If it starts to brown too quickly, loosely tent it with a piece of aluminum foil and continue baking until done.
16. Cool and Serve: Let the bread cool completely at room temperature before slicing and enjoying this delicious, healthy keto brioche!