Put the both flours into a large mixing bowl and stir in the salt, yeast and caraway seeds. Make a well in the centre.
Lightly whisk 2 of the eggs with the treacle or molasses 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 20cm (8 in) long log, then shape it into a ring. Dampen the ends with a little water, slightly overlap them and gently pinch together to seal.
Arrange the bagels on a lightly oiled baking tray, cover with oiled cling film and leave to rise in a warm place for 20 minutes or until they are slightly puffy.
Preheat the oven to 200 C / Gas 6. Bring a large pan of lightly salted water to the boil. Drop the bagels into the water, one at a time, and poach for 20 seconds. Lift out with a large draining spoon and return to the baking tray. Lightly beat the remaining egg and brush it over the bagels.
Bake until well risen and golden brown, about 14 minutes. Transfer to a wire rack to cool. The bagels can be kept in an airtight container for up to 3 days. They also freeze well.