Dissolve the yeast in the lukewarm water in a large bowl. Pour the levain into the large bowl along with the sugar, both flours and the salt. Mix to form a rough dough then tip the contents of the bowl out onto a clean worksurface. Knead for 10 minutes until smooth and elastic, dusting your hands and the dough with flour as needed to stop it sticking to the surface. Pour a little vegetable oil into the large bowl you were using earlier, place the dough into it and turn to coat. Cover with cling film and leave to rise in a warm place for 4 hours.
Punch down the dough, divide it into 8 pieces and roll each piece into a ball. Poke a hole in the centre of each ball and stretch it into a hoop - you want the hole to be about 2-3cm wide. Place on an oiled baking tray and cover with oiled cling film then leave in the fridge overnight.
In the morning, bring a large pot of water to the boil and preheat the oven to 200 C (400 F).
Once the water is boiling, stir in the sugar and baking soda and turn the heat down to simmer. Gently lower in a bagel - it’ll sink at first and then float to the surface. Add another bagel to the pot if it'll fit. Let the bagels simmer in the pot for two minutes then flip and boil for another 2 minutes on the other side.
Remove the boiled bagels from the pot using a slotted spoon and place back onto the greased baking tray. Immediately sprinkle with sesame seeds.
Repeat the boiling & sesame seed sprinkling with the rest of the shaped bagels. You can arrange them so they’re almost touching on the baking tray as they won’t really rise in the oven.
Bake for 20-22 minutes until deep golden. Leave to cool for 10 minutes before slicing in half and eating. They'll keep in a plastic bag for 2-3 days.
(Tip: if you know you won't eat the bagels within 2 days, slice them and place into a ziplock bag to freeze. You can toast them straight from frozen)