Preheat the oven to 325°F (163°C). Adjust the oven rack to the middle position.

In a Dutch oven or flameproof casserole dish over medium heat, warm 1 to 2 tablespoons oil. Add the onion and carrot and cook, stirring occasionally, until they begin to soften, 5 to 7 minutes.

Season the beef with salt and pepper. Place the flour in a shallow bowl and then roll the meat in the flour. Add the beef to the vegetables and cook, turning once, just until browned, adding more oil if necessary, 5 to 10 minutes.

Stir in the tomato paste. Pour in the beer and use a wooden spoon to stir and scrape up any browned bits from the bottom of the Dutch oven.

Bring to a boil and then add the slice of bread with the mustard. Stir until the bread falls apart, 2 to 5 minutes.

Cover and place in the oven. Cook until the beef is tender, 1 ½ to 2 hours. (Alternatively, you can very gently simmer the stew on the stovetop so that it's only barely simmering.) Check on the stew occasionally. If it is looking dry, pour in up to 1 cup of water. Let the filling cool while you make the pastry.

In a food processor fitted with the blade attachment, combine the flour, salt, butter, and cheese. Pulse until the mixture resembles bread crumbs, about 8 seconds.

Add the water and pulse until the dough forms a ball in the bowl. If the dough is dry, drizzle in more water, 1 teaspoon at a time. Remove from the bowl and knead briefly, just to bring the pastry together, drizzling in additional water, if necessary.

Wrap the pastry in plastic wrap and let it rest in the refrigerator for 30 minutes.

Preheat 350°F (177°C) and butter the pie pans.

Set aside ⅓ of the pastry for the lids. Roll out the remaining pastry on a floured work surface until ⅛ inch (3 mm) thick. Roughly cut the pastry into 4 or 6 pieces, depending on the size and number of pie pans you are using. Each piece should be slightly larger than the size of the pans. Gently lift each piece of pastry over a pan and let it sink into the base. Let the excess dough hang over the edge to help attach the pastry lid. Gently use your fingers to press the edges into the tins.

Roll out the pastry for the lids until large enough to cover the pies and fill the pies with the cooled stew.

Place a piece of pastry on top of each pie and use a sharp knife to cut the excess pastry from the base and top. Squeeze the edges together so that the pies don't open during baking.

Brush the top of each pie with the egg wash. Bake until the pies are golden brown, 45 to 65 minutes. You can also freeze the unbaked pies and bake them straight from the freezer. Let cool slightly before serving.