Place the flours in a large mixing bowl and stir in the salt, yeast and caraway seeds. Make a well in the centre.
Lightly whisk 2 eggs with the treacle and oil, and pour into the well in the flour. Add the water and mix to a soft dough.
Turn out onto a lightly floured surface and knead for 10 minutes or until smooth and elastic. Place the dough in a large greased bowl, cover with a damp tea-towel and leave to rise in a warm place for 40 minutes or until doubled in size.
Turn out the dough onto the floured work surface and knead it lightly, then divide it into 12 equal pieces. Form each into a 20 cm long sausage, then shape it into a ring. Dampen the ends with a little water, slightly overlap them and then gently pinch together to seal.
Arrange the bagels on a lightly oiled baking tray, cover with oiled plastic wrap and leave to rise in a warm place for 20 minutes or until they are slightly puffy.
Preheat the oven to 200 degrees C. Bring a large saucepan of lightly salted water to the boil. Drop the bagels into the water, one at a time, and poach for 20 seconds. Lift them out with a large slotted spoon and return to the baking tray.
Lightly beat the remaining egg and brush it over the bagels to glaze. Bake for 14–15 minutes or until well risen and golden-brown. Transfer to a wire rack to cool. The bagels can be kept in an airtight container for up to 3 days.