Gently warm the milk until tepid (not hot).
Stir in the yeast and 1 tsp (5 ml) of the sugar.
Cover and set aside in a warm place for about 10 minutes until it becomes frothy.
In the bowl of a stand mixer fitted with a dough hook, combine the flour, the remaining 45g (1.59 oz) of sugar, and the salt. Make a well in the center.
Pour the frothy yeast mixture and the beaten eggs into the well.
Knead on a low speed for about 7 minutes until a shaggy dough forms.
With the mixer still running on low, begin adding the softened butter one heaped tablespoon at a time. Wait until each piece is fully incorporated before adding the next. This is a slow process but crucial for the texture. Scrape down the sides of the bowl as needed.
Once all the butter is incorporated, cover the bowl with cling film and leave it to prove in a warm, draught-free place for 45 minutes.
Gently knead the dough for 3 minutes to knock it back.
Place it back in the bowl, cover, and let it prove for a second time for 30 minutes.
Knock the dough back one last time.
Divide it into 12 equal pieces.
Shape each piece into a ball and place it into a floured muffin mould (silicone is great, otherwise line with paper cases).
Cover the moulds loosely with cling film and let them prove for a final 20 minutes, or until they have increased in size by about one-third.
Preheat your oven to 175°C (350°F) / 155°C Fan / Gas Mark 4.
Bake the buns for 10 minutes.
Remove them from the oven, quickly brush the tops with the beaten egg for the glaze, and sprinkle with a little caster sugar.
Return the buns to the oven and bake for a final 3-5 minutes, or until they are a beautiful golden brown.
Let the buns cool in their moulds for 10 minutes before transferring to a wire rack to cool further.
Serve warm!
Patience with the Mixer: Don't rush adding the butter. Incorporating it slowly, piece by piece, is the most important step for developing the dough's strong, silky structure.