Preheat the oven to 180C/350F/Gas 4.
Whisk one of the eggs in a bowl. Beat in the flour, cheese, butter, one tablespoon of the caraway seeds and a pinch of salt until the mixture comes together as a dough. Set aside to rest for 30 minutes.
Roll the dough out on a floured surface to 1cm/½in thickness. Using a pretzel cutter or a biscuit cutter, cut the pretzels out and place onto a baking tray lined with greaseproof paper. (you may need more than one baking tray.)
Beat the remaining egg in a bowl and brush over pretzels. Sprinkle the remaining caraway seeds over the pretzels and the extra grated cheese.
Bake in the oven for 10-15 minutes, or until golden-brown. Remove from the oven and set aside to cool on a wire rack.