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In a glass measuring cup, combine one cup warm water, the milk, yeast and sugar. Let stand until foamy, about five minutes. Meanwhile, beat one egg.
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In the bowl of a food processor fitted with a dough blade, add both flours, salt and butter. Pulse several times until the butter is blended in and the flour mixture resembles sand. Beat the egg and add to the flour mixture.
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With the food processor running, slowly pour in water mixture and mixing until the dough forms a ball.
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Lightly butter a large bowl. Shape dough into a ball and place it in the buttered bowl. Cover bowl with plastic wrap and let rise in a warm place until doubled in bulk, one to two hours.
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Line a baking sheet with parchment paper. Using dough scraper, divide dough into 8 equal parts. Gently roll each into a ball, flatten slightly and shape into a disk. Arrange two to three inches apart on baking sheet. Cover loosely with a clean dish towel and allow to rise again for 1 to 1 1/2 hours or until the dough feels like a marshmallow when gently touched.
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Preheat oven to 400 degrees with rack in center. Beat remaining egg with one tablespoon water and brush the tops and around the sides of the bun. Sprinkle with sesame seeds, if desired. Bake baking, until tops are golden brown, about 15 minutes. Transfer to a rack to cool completely. Makes 8 4 to 5-inch burger bun.