Activate the yeast. In a small bowl, combine the active dry yeast with 1–2 tablespoons of lukewarm milk and a pinch of sugar. Stir gently and let it sit for about 10 minutes, until slightly foamy.
Make the dough. In a large mixing bowl or the bowl of a stand mixer, combine the flour, sugar, and salt. Make a well in the center, then add the egg, activated yeast mixture, and remaining milk.
Knead until smooth. Mix everything together, then knead for about 20 minutes by hand or 10–15 minutes with a stand mixer. The dough should become smooth, elastic, and slightly sticky. Add the softened butter, a little at a time, and continue kneading until completely incorporated.
Add the chocolate chips. If the dough feels warm after kneading, let it rest for a few minutes. Then gently fold in the chocolate chips, making sure they are evenly distributed without overworking the dough.
Let the dough rise. Cover the bowl with a clean kitchen towel and leave the dough at room temperature for about 1 hour, or until it has doubled in size.
Shape the buns. Divide the chilled dough into 2.3-ounce (65g) portions. Shape each piece into a smooth ball on a lightly floured surface. Work gently and avoid overhandling the dough, especially if your chocolate chips start to soften.
Let them rise again. Arrange the buns on a lined baking sheet, leaving some space between them. Brush lightly with a mixture of egg yolk and milk, then cover and let them rise for about 1 hour, until noticeably puffy.
Bake until golden. Preheat the oven to 350°F (180°C). Brush the brioche buns once more with the egg yolk and milk mixture, then bake for 15–20 minutes, or until beautifully golden brown. Transfer them to a wire rack and let them cool before serving—although I’ll admit, they’re very hard to resist while still warm!