In a large bowl, whisk together the flours and salt. Scatter the butter pieces over the top, and work with a pastry blender, or your fingers, to combine, until the mixture resembles gravel, with lots of butter chunks the size of large peas.
Add the lightly beaten egg, and stir into flour mixture with a wooden spoon. Drizzle the vodka into the flour mixture, then the water, 1 Tablespoon at a time, tossing the dough with a rubber spatula to moisten evenly. Add just enough liquid for the dough to hold together when you give it a squeeze, and add it directly to the dry floury bits that like to hang out on the bottom of the bowl.
Form the dough into a ball, wrap it in plastic or beeswax wrap, and place it in the fridge for 1 hour to firm. Meanwhile, prepare the filling.
Combine the goat cheese, egg yolks, green onions and dill in a food processor, and pulse until smooth. Keep chilled until ready to use.
Preheat the oven to 400°F.
Remove the dough from the fridge, and use a bench scraper or a sharp knife to divide the dough evenly into 6 pieces. Roll each piece into a ball, being careful not to overwork the dough. On a lightly floured surface, roll each ball into a 6 to 7-inch circle.
Spoon 1-2 Tablespoons of the filling into the center of the circle. Spread the filling into an even layer, leaving a one-inch border.
Arrange three tomato slices, or an array of cherry tomatoes, evenly over the center of the filling. Fold the dough over the tomatoes in a free-form fashion, pleating the overlaps.
Brush the outer crust with a generous coating of the beaten egg wash, and sprinkle with the Everything Bagel Seasoning.
Bake for 20 minutes, or until golden brown. Let cool before serving. Garnish with a drizzle of balsamic reduction, a spoonful of capers, and a sprig of fresh dill. Enjoy!