In a small bowl or liquid measuring cup, sprinkle yeast over 1/2 cup water. Add 2 teaspoons honey. Whisk until yeast dissolves. Let stand until foamy, about 5 minutes.
Transfer to the bowl of a mixer fitted with the dough hook. Add butter and remaining 1 3/4 cups plus 2 tablespoons water and 3 tablespoons honey. Whisk together all-purpose, whole-wheat, and rye flours with salt; add 3 cups to yeast mixture. Mix on low speed until smooth. Add remaining 3 1/4 cups flour mixture, 1 cup at a time. Add wheat berries, bulgur, 1/2 cup oats, 1/2 cup flaxseed, and 1/3 cup sunflower seeds; mix on low to combine. Continue mixing until dough comes away from sides of bowl and forms a ragged, slightly sticky ball.
Butter a large bowl. Knead dough on a floured surface until smooth and elastic but still slightly tacky, about 5 minutes. Shape into a ball. Transfer to prepared bowl; cover with plastic wrap.
Let dough stand in a warm place until it doubled in volume (it should not spring back when pressed), about 1 hour. Butter two 4 1/2-by-8 1/2-inch loaf pans. Punch down dough and divide in half.
Working with one piece of dough at a time, shape into a 7-by-7-inch square. Fold dough into thirds; press seam to adhere and pinch ends to seal. Place seam side down in loaf pan. Repeat process with remaining piece of dough.
In a small bowl, mix egg white with 1 teaspoon water and brush mixture over tops of loaves; sprinkle with remaining 2 tablespoons rolled oats, remaining 1 tablespoon flaxseeds, and remaining 2 tablespoons sunflower seeds.
Spray two pieces of plastic wrap with nonstick cooking spray; drape loaves with prepared plastic wrap and let stand until dough rises about 1 inch above tops of pans, 45 minutes to 1 hour.
Preheat oven to 450 degrees for conventional, 425 degrees for convection. Reduce oven temperature to 400 degrees (conventional) or 375 degrees (convection).
Uncover loaves and transfer to oven. Bake, rotating pans after 20 minutes, until tops are golden brown and internal temperature reaches 205 degrees on an instant-read thermometer, about 45 minutes total. Transfer to wire racks. Let cool slightly; turn out loaves. Let cool completely before slicing.