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1
In a large bowl, whisk together the flours and salt. Using a pastry cutter or knife and fork, cut in the butter until it is the texture of cornmeal and peas.
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2
Make a well in the center of the butter-flour mixture and pour in the beer. Using a wooden spoon, combine until the dough forms together into a flat ball (you may need to use your hands at the end). Fold the dough over itself and wrap in plastic wrap, then let chill in the fridge for about 30 minutes.
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3
Preheat the oven to 375u00b0 F. Add 1 tablespoon olive oil and sliced onions to a cast iron skillet and place over medium heat. Stir to coat the onions with olive oil. Cook the onions, stirring occasionally, until translucent. Sprinkle them with salt and pepper and reduce heat to medium-low. Cook 25 to 30 more minutes, until the onions are caramelized.
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4
While the onions are cooking, prepare the rest of the galette. In a small bowl, whisk together the balsamic vinegar and mustard and set aside.
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5
After 30 minutes, remove dough from the fridge and unwrap. On a floured surface, roll it out into an elongated rectangle. Pick up the bottom of the rectangle, and fold the dough 2/3 of the way up. Now pick up the top third of the dough and fold it over the bottom. Sprinkle more flour over the dough, rotate it 90 degrees, and then repeat the same folding technique.
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6
Roll out the dough into a 10- or 11-inch circle on a sheet of parchment paper. Transfer the parchment and dough to a large cookie sheet.
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7
On top of the crust, brush on the mustard-vinegar mixture and spread evenly. Add the cheese, ham, and caramelized onions, scattering them evenly across the crust, but leaving a 1-inch border. Fold the edges over the top of the filling and press to seal. Brush olive oil on the crust edges and sprinkle the entire tart with sea salt and pepper.
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8
Place a sheet of aluminum foil over the filling, leaving the crust exposed (this will keep the filling from browning too quickly or burning). Bake for 35 to 50 minutes, until the crust is browned. Remove from the oven and cool on a rack while you fry the egg.
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9
Heat a pat of butter in a small skillet. Fry the egg, sunny-side up, until the white is no longer translucent and the edges have crisped. Using a skillet, transfer the egg to the tart. Serve immediately and enjoy with a glass of hard, dry cider -- I recommend Foggy Ridge First Fruit!
