Make the dough: Pulse the flour and fine salt in a food processor to combine. Add the butter and cheese and pulse until the mixture looks like coarse meal with some pea-size bits of butter. Drizzle in 6 tablespoons beer and pulse to combine. Squeeze the dough between your fingers; if it does not hold its shape, pulse in up to 1 more tablespoon beer. Pat the dough into a disk, wrap in plastic wrap and refrigerate until firm, at least 1 hour or overnight.
Meanwhile, make the filling: Preheat the oven to 325 degrees F. Toss the beef, flour, 1 1/4 teaspoons kosher salt, and pepper to taste in a large bowl. Melt the butter in a Dutch oven or large ovenproof pot over medium-high heat. Add the beef and cook, stirring occasionally, until browned all over, about 7 minutes. Add the beer and 2 cups water, scraping up any browned bits from the bottom of the pot. Bring to a simmer, then cover, transfer to the oven and braise, 30 minutes. Add the leeks, carrots and celery to the pot and continue braising, 1 more hour.
Meanwhile, put six 10-ounce ramekins on a parchment-lined baking sheet. Roll out the dough between 2 lightly floured sheets of parchment into a 10-by-15-inch rectangle; cut into six 5-inch squares. Make a few slits in the middle of each square to let steam escape. Remove the filling from the oven and increase the temperature to 375 degrees F. Season the filling with salt and pepper and divide among the ramekins (there may be some filling left over). Top each with a square of dough, pressing the edges into the ramekin. Brush with the beaten egg. Bake until the crust is crisp and golden brown, 35 to 40 minutes. Let rest about 10 minutes before serving. Photograph by Anna Williams