Preheat the oven to 180°C, oil a large bowl and grease two large baking sheets.
Mix flour, yeast, caster sugar, cayenne pepper, and dried oregano in a large bowl.
Combine warm water with salt and oil in a separate bowl, then slowly mix into the dry ingredients.
Knead the dough on a floured surface until smooth and elastic.
Place dough in the oiled bowl, cover with a damp cloth, and rest for 10 minutes.
Knead briefly for one minute, cover, and leave to prove for another 10 minutes.
Incorporate cheese and chillies into the dough and prove for 30 minutes.
Divide dough into 12 balls and prove covered for 20 minutes.
Shape balls into rings by stretching the center with your fingers.
Boil water with brown sugar and poach bagels for 30 seconds per side.
Transfer bagels to baking trays using a slotted spoon.
Brush with egg, sprinkle with extra seasonings, and bake for 20-25 minutes.