In a small bowl, combine the oats and milk and let stand until the oats soften slightly, about 5 minutes.
In a food processor, combine the flour, baking powder and salt and pulse a few times to blend. Add the cheese and butter and pulse until a coarse meal forms. Stir the beaten egg into the softened oats, then scrape the oats into the food processor. Pulse until a dough forms. Scrape the dough onto a large sheet of plastic wrap and gently knead a few times until thoroughly blended. Pat the dough into a disk, wrap it up and refrigerate until firm, about 1 hour.
Preheat the oven to 375°. Line 2 baking sheets with parchment paper. Work with half of the dough at a time: On a lightly floured work surface, dust the dough with flour. Cover with a large sheet of plastic wrap. Roll out the dough 1/8 inch thick. Quickly cut the dough into 1 1/2-inch squares or stamp out different shapes. Brush off any excess flour and transfer the squares to one of the baking sheets. Refrigerate for at least 5 minutes, until the squares are firm. Repeat with the second piece of dough.
Lightly brush the squares with the egg wash and lightly sprinkle with sugar. Bake for about 16 minutes, until the crackers are golden brown. Transfer to a rack and let cool before serving or packaging.