Set aside 6 tablespoons of flour from the total amount of flour. Mix the yeast and remaining flour in the bowl of your food processor. Add the warmed milk, beaten egg, sugar and a pinch of salt.

Continue mixing, then gradually add the soft butter pieces. Let the processor run for 5 minutes at low speed, then 5 minutes at slightly higher speed.

It's sticky, but that's normal. Place the dough on a generously floured work surface, gradually add the remaining flour (a little more if necessary, until the dough forms a much less sticky ball) and knead with your hands.

Tip to determine whether the dough has been kneaded enough: press lightly with your finger. If the dough more or less returns to its original shape (but not completely), it's ready! Cover and leave to rise at 82-86°F/28-30°C (see notes) for 1h30.

Flour a large sheet of parchment paper and place the dough on top. Roll out with a well-floured rolling pin, then spread with Pesto. Grate the cheeses and add them.

Roll up the dough, then cut the sausage in half from bottom to top. Braid the dough, placing one end on top of the other. If you find it too hard to braid, place the braids in the freezer for 15 minutes.

Beat an egg yolk and 2 tablespoons of milk in a bowl, and brush gently with the mixture. Set the mixture aside in the fridge and leave to rise again, covered, at 82-86°F/28-30°C for 45 minutes.

At the end of the rising time, preheat the oven to 340°F/170°C on natural convection/static heat high and low, and brush the braid lightly and very gently with the mixture.

Bake for 40-45 minutes in the middle of the oven. Keep a close eye on the Babka as it cooks: if it darkens too much and there's still time to bake, cover it with aluminum foil.