To make the brioche, sprinkle the yeast and caster sugar on to the warm milk in a small bowl. Whisk them together lightly, then leave in a warm place for 10 minutes or until a light foam forms on the surface.
Sift the flour, salt and caster sugar into a large mixing bowl. Make a well in the centre and pour in the yeast mixture and the beaten eggs. Mix together with a wooden spoon until the mixture forms a ball.
Turn the dough out on to a floured surface and knead it gently for about 5 minutes until smooth. Gradually add the softened butter, a teaspoon at a time, kneading well between each addition. It will take about 10 minutes to incorporate the butter and the dough will be very sticky.
On a clean floured surface and using clean hands, sprinkle the dough with flour and continue to knead lightly for 10 minutes until the dough is very pliable, smooth and no longer sticky. Place it in a large buttered bowl and cover with a clean cloth. Leave in a warm place for 1½ hours until well risen and spongy.
Turn the dough back on to a lightly floured surface and knock it back with your knuckles.
Knead for a minute or so, then flatten the dough slightly and wrap it around the saucisson, pressing the edges firmly to seal.
Place it in a lightly buttered 900ml/1½ pint loaf tin and cover loosely with cling film. Leave in a warm place to prove for 45 minutes or until the dough is well risen.
Preheat the oven to 180C/350F/Gas 4.
Brush the brioche lightly but generously with beaten egg yolk and bake for 35–40 minutes until risen and golden brown.
Cool it in the tin for 10 minutes, then turn out and cool on a wire rack.
Serve in thick slices.