Preheat the oven to 350°F (180°C). Line a baking sheet with parchment paper.
Place the flour on a large cutting board or clean work surface (something you can safely cut on with a sharp knife). Scatter the cold, cubed butter over the flour.
Using a long-bladed knife, chop and slice through the butter and flour in different directions until the mixture resembles coarse breadcrumbs. This traditional Lithuanian technique replaces rubbing the butter in by hand and keeps the pastry tender.
Spread the grated cheese over the flour mixture. Add the sour cream in small spoonfuls across the surface. Using the same knife, continue chopping and slicing until the dough begins to come together.
If the mixture feels too dry, add a little more sour cream, 1 teaspoon at a time.
Gather the mixture into a ball and knead gently just until it holds together. Flatten into a rough rectangle, wrap in plastic wrap, and refrigerate for 30 minutes.
Lightly crush the caraway seeds in a mortar and pestle, or roughly chop them with a knife to release more flavor.
Roll the chilled dough into a rectangle about 5 mm (⅕ inch) thick (ours measured 7cm (3in) wide and 30 cm (12in) long).
Using a pizza cutter or sharp knife, cut into strips approximately ⅓ inch (1 cm) wide and about 3 inches long - we tycally end up with 30 sticks.
Without moving anthing, brush the surface wit hwith the beaten egg. Sprinkle with crushed caraway seeds and a light pinch of salt - it's quicker than topping them individually.
Place the sticks on the prepared baking sheet, leaving a little space between them.
Bake in the center of the oven for 13–15 minutes, or until golden brown on top.
Allow the cheese straws to cool slightly on the baking sheet. They will firm up as they cool.