all-purpose flour, plus more for rolling
large pinch of salt
stick cold unsalted butter, diced
sharp cheddar cheese, shredded (1 cup)
plus 1 teaspoon milk
In a food processor, pulse the 1/2 cup of flour with the cardamom and salt. Add the butter and pulse until it is the size of small peas. Add the cheddar and 1 teaspoon of the milk and pulse just until the dough comes together. Scrape the dough onto a work surface and knead a few times until smooth. Form the dough into a disk, wrap it in plastic and refrigerate until firm, about 30 minutes.
Preheat the oven to 350°. Line two large rimmed baking sheets with parchment paper. In a small bowl, beat the egg with the remaining 1 tablespoon of milk. On a lightly floured work surface, roll out the dough to a 10-by-9-inch rectangle, 1/4 inch thick. Cut the dough lengthwise into 1/4-inch-wide straws.
Place the straws on the prepared baking sheets, about 1/2 inch apart. Brush the straws with the egg wash. Bake for about 15 minutes, until golden brown, rotating the pans halfway through. Let cool on the baking sheets until crisp, about 10 minutes, before serving.
Make Ahead: The cheese straw dough can be wrapped in plastic and refrigerated for 2 days. The cheese straws can be kept in an airtight container for up to 2 days.