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In a medium bowl, mix the sourdough starter, water, eggs, melted butter, and honey.
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Add the whole wheat flour and salt. Mix well.
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Add the all-purpose flour ¼ cup at a time, until your dough comes together.
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Knead the dough for 10-15 minutes by hand. (This dough can also be made in your stand mixer or in your bread machine on the dough setting.) Once kneaded, the dough should be soft and slightly sticky.
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Place kneaded dough in a clean, greased bowl and cover it with a damp tea towel. Place the bread in a warm, draft-free place to rise for 4 hours. (If you want a more sour bun, you can let the dough rise for 6-8 hours.)
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After dough has risen, punch it down and turn it out onto a lightly floured surface. Divide the dough into 10 roughly equal pieces. Roll each piece into a ball and flatten the ball to about 1-inch thick. Place the rounds on a greased baking sheet.
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Cover the shaped buns with your tea towel and let them rise for another 1-2 hours, until nearly doubled and puffy.
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Near the end of the rising time, preheat your oven to 400°F.
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Once risen, gently brush the buns with with the egg wash and sprinkle with sesame seeds (if desired).
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Bake the buns in your preheated oven for 15-20 minutes, until golden and firm when tapped.
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Transfer the buns to a wire rack and let them cool completely before using or storing. Slice before using.