This recipe, created by Rosemary Leicht of Bethel, Ohio, was named grand champion at the 2013 National Festival of Breads. While Rosemary calls for the dough to be made in a food processor, it's fine to make it by hand or in a stand mixer, instead; see "tips," below.
Place the kneading blade in an 8-cup food processor bowl. Add 1/3 cup warm water, yeast and honey; pulse a few times. Let stand 10 minutes.
Add the flours, olive oil, salt, 1/3 cup water, and onion. Pulse 3 to 4 times to mix; process 1 minute. The dough should form a ball around the blade. If additional water or flour is needed, add by the tablespoon.
Place the dough in a lightly greased bowl, turning to grease the surface. Cover and let rise in a warm place for 45 minutes.
Arrange the oven racks so one's at the top, and one's at the bottom. Preheat the oven to 400°F.
Divide the dough into four equal balls. Working with one ball at a time, roll the dough into a 12" to 13" circle. Place each circle on a large piece of parchment paper.
Roll the remaining dough into 12" to 13" circles, placing each on a sheet of parchment paper. Dust any remaining flour off the circles with a pastry brush; if you don't, the surface will become gummy when you brush on the egg white, reducing the crispness.
Allow the circles to rest for 5 minutes. Brush some of the beaten egg white on each circle; sprinkle each with ¼ of the cheese.
Slide one dough circle with its parchment onto a large baking sheet; repeat with a second dough circle on another baking sheet.
Place one baking sheet on the oven's top rack, the other baking sheet on the bottom rack.
Bake the breads for 4 to 5 minutes, then reverse the baking sheets top to bottom and bake for another 4 to 5 minutes. The dough should puff up in places, and turn golden.
Remove the baking sheets from the oven, and slide the breads off the parchment onto a rack to cool.
Repeat with the remaining two pieces of dough. Note: These last two breads may bake more quickly, as the sheets are hot; so watch carefully.
Break the cracker breads into pieces to serve; or place in an airtight container to store.
