In a large bowl, mix warm water, yeast, sugar, and honey. Let stand until foamy, 5-10 minutes.
To the yeast mixture, add olive oil, salt, and 2 cups flour. Stir in the remaining flour, 1/2 cup at a time, until the dough holds together.
Turn out onto a lightly floured surface, and knead until smooth and elastic, about 8 minutes. (Dough will be loose and sticky, but not too sticky).
Lightly oil a large bowl, place the dough in the bowl, and turn to coat. Cover with a damp cloth, and let rise in a warm place until doubled, about 1 hour.
Preheat oven to 400 degrees.
Deflate the dough, and turn it out onto a lightly floured surface. Divide the dough into 16 equal pieces, (I cut it like a pizza, halved then quartered and so on, so they all come out equal) and form into round balls. Place on lightly greased baking sheets at least 2 inches apart. Cover the rolls with a damp cloth, and let rise until doubled again, about 40 minutes.
Brush the tops with egg white. (Feel free to use your fingers, just don't press too hard).