In the bowl of a stand mixer fitted with the dough hook, combine the all-purpose flour (500g / 4 cups), granulated sugar (75g / 1/3 cup + 1 tbsp), salt (10g / 2 tsp), and instant dry yeast (7g / 2 1/4 tsp). Whisk briefly to distribute the ingredients evenly.

In a separate small bowl, whisk together the lukewarm whole milk (150ml / 2/3 cup) and the room temperature eggs (3 large, about 150g). Ensure the milk is not too hot, as it can kill the yeast; it should feel comfortably warm to the touch.

Pour the milk and egg mixture into the dry ingredients in the stand mixer bowl. Mix on low speed until a shaggy dough forms and all the dry ingredients are incorporated. This should take about 1-2 minutes.

Increase the mixer speed to medium-low and knead the dough for about 8-10 minutes. The dough will start to become smoother and slightly elastic. It will still be quite sticky at this stage.

With the mixer still running on medium-low, gradually add the softened, cubed unsalted butter (170g / 3/4 cup) one cube at a time. Allow each addition of butter to be mostly incorporated before adding the next. This process will take about 5-7 minutes.

Once all the butter is added, increase the mixer speed to medium and knead for another 8-10 minutes. The dough should become very smooth, elastic, and slightly shiny. It will still be quite soft and a bit sticky, but it should pull away from the sides of the bowl cleanly.

Lightly grease a clean large bowl with a neutral oil or cooking spray. Transfer the brioche dough to the prepared bowl, turning it once to coat the entire surface. Cover the bowl tightly with plastic wrap or a damp kitchen towel.

Let the dough rise in a warm place for 1.5 to 2 hours, or until it has doubled in size. A good warm spot is an oven that has been turned off after being preheated briefly to around 38°C (100°F).

Gently punch down the dough to release the air. Turn the dough out onto a lightly floured surface. Shape it into your desired loaf form. For a standard loaf pan (approximately 23cm x 13cm or 9x5 inches), you can shape it into a log and place it seam-side down in the greased pan. Alternatively, you can divide the dough into smaller portions and arrange them in the pan for a more voluminous loaf.

Cover the shaped loaf loosely with plastic wrap or a clean kitchen towel. Let it rise again in a warm place for another 1 to 1.5 hours, or until it has nearly doubled in size and looks puffy. It should rise about 1.5-2.5cm (0.5-1 inch) above the rim of the pan.

Preheat your oven to 190°C (375°F). Once the dough has completed its second rise, gently brush the top of the loaf with the prepared egg wash (1 beaten egg with 1 tbsp milk or water). Be careful not to deflate the dough.

Bake the brioche loaf in the preheated oven for 30-40 minutes. If the top starts to brown too quickly, you can loosely tent it with aluminum foil for the remainder of the baking time. The bread is done when it's golden brown and sounds hollow when tapped on the bottom.

Remove the brioche loaf from the oven and immediately turn it out of the pan onto a wire rack. Let it cool completely before slicing, at least 1-2 hours. This allows the internal structure to set properly.