Soak the Bulgur: Place the bulgur wheat in ¼ cup warm water and let it stand for 30 minutes, or until the water has been completely absorbed.
Warm the Liquid Mixture: Combine the remaining 1 cup water, honey, and unsalted butter in a small saucepan. Warm over medium heat until the butter has melted. Remove from the heat and cool for about 5 minutes, until the mixture reaches approximately 120–130°F.
Combine the Dry Ingredients: In the bowl of a stand mixer fitted with a dough hook, add the active-dry yeast, whole wheat flour, rye flour, and coarse salt.
Form the Dough: Pour the warm water mixture into the dry ingredients and mix until combined. Add the rolled oats, flaxseed, sunflower seeds, and soaked bulgur, then mix again until the ingredients are evenly incorporated.
Add the All-Purpose Flour: Add the all-purpose flour a few tablespoons at a time until the dough clears the sides of the bowl. You may not need the full 1½ cups.
Knead the Dough: Continue kneading with the mixer for about 3–4 minutes, until the dough becomes smooth and elastic. The dough will be heavier than many bread doughs, but it should stretch slightly without immediately breaking.
First Rise: Shape the dough into a ball and place it in a greased bowl. Cover with plastic wrap and set it in a warm place. Let it rise for about 1 hour, or until doubled in size and the dough does not spring back when gently pressed.
Rest the Dough: Punch down the risen dough and knead it a few times until smooth. Allow it to rest for about 5 minutes.
Shape the Loaf: Line a Dutch oven or large heavy pot with parchment paper and place the dough inside.
Add the Topping: If desired, brush the surface with the beaten egg wash and sprinkle additional rolled oats over the top.
Score the Bread: Using a sharp knife, make 3 cuts across the top of the loaf.
Second Rise: Cover the loaf with plastic wrap and let it rise for another 30–45 minutes.
Preheat the Oven: Meanwhile, preheat the oven to 425°F and position the rack in the center of the oven, or lower if needed to accommodate the pot.
Bake the Bread: Cover the Dutch oven with its lid and place it in the preheated oven. Immediately reduce the oven temperature to 400°F and bake for about 45 minutes, until the top of the bread is golden brown.
Cool Before Slicing: Remove the bread from the oven and allow it to cool before slicing. For the best eating quality, enjoy it on the day it is baked; it can also be stored in an airtight container at room temperature for several days.