Puree the beetroot with the egg and put into a mixing bowl.
Add the remaining ingredients for the dough and knead into a smooth dough.
Cover and leave to rest in a warm place for about 1 hour.
Chop the couverture.
Heat the butter and sugar in a saucepan, stirring constantly.
Remove from the heat and stir-in the couverture until it dissolves.
Mix in the cocoa powder to make a spreadable mixture.
Transfer the dough to a well-floured work surface.
Roll out.
Spread the filling evenly.
Roll up.
Cut the roll lengthwise.
Twist the two strands into each other so that the opening faces up.
Place in a greased and floured loaf tin (30 cm).
Leave to rise again for ½ an hour.
Bake in preheated oven at 180°C for about 40 minutes. After about 20 minutes, cover with aluminium foil to prevent the babka from getting too dark.
Mix the icing sugar and milk. If necessary, heat briefly in the microwave to make the glaze more liquid.
Spread on the beetroot babka.
Leave to cool on a rack.