Preheat the oven to 180C/350F/Gas 4, lightly oil a large bowl and grease two large baking sheets.
In a large clean bowl, mix the flour, yeast, caster sugar, cayenne pepper and dried oregano together. In a separate bowl, mix the warm water, salt and oil, then slowly combine with the dry ingredients.
On a floured surface, knead the dough until smooth and elastic. Transfer to the oiled bowl, cover with a damp cloth and leave for 10 minutes. Briefly knead the dough again for a minute, cover again and leave to prove for a further 10 minutes.
Add the cheese and chillies to the dough and knead until evenly distributed. Cover and set aside to prove for 30 minutes.
Divide the dough into 12 equal pieces and roll each piece into a neat ball. Cover with a damp tea towel and set aside to prove for 20 minutes.
Shape the dough balls into rings by inserting a wooden spoon into the centre and using your fingers to gently stretch the dough.
To prepare the poaching water, fill two large pans with water, add the brown sugar and bring to the boil. Poach the bagels two or three at a time for 30 seconds on each side. Lift them out with a slotted spoon, shake off any excess water and lay them on the prepared baking trays.
Brush lightly with egg, sprinkle over the oregano and smoked paprika then transfer to the oven for 20-25 minutes until golden-brown.