Position a rack in the center of the oven and preheat it to 425F.

Peel and wrap the beet in aluminum foil. Roast until soft, 35 to 45 minutes.

Slice the beet and add it to a blender with ⅓ cup / 80g cool water. Blend until smooth. This should yield ¾ cup / 140g beet puree; discard any extra.

In a small measuring cup, combine the melted butter and warm water. Whisk in the sugar and let it dissolve. Ensure the entire mixture is between 100 and 115F before sprinkling in the yeast. Stir to combine and let sit until the yeast is foamy, about 5 minutes.

In a stand mixer fitted with the dough hook, combine the flour, salt, and beet puree. Pour in the yeast mixture and mix on medium-low speed until everything is just combined. Increase the speed to medium and mix until the dough starts to pull away from the sides of the bowl, 7 to 8 minutes. Use a damp hand to pull the dough from the hook and transfer it to an oiled bowl. Cover and let this proof until at least doubled in size, about 1 ½ hours.

Line the bottom and sides of a 9 by 5-inch loaf pan with parchment paper.

When the dough has doubled in size, punch it down and transfer it to a lightly floured surface. Gently pull the dough out into a rectangle and fold one side two-thirds of the way over, fold the other side over the top of that side, then roll the dough into a log, starting from the top narrow edge and rolling down. Transfer this, seam-side down, to the lined loaf pan. Cover it loosely with plastic wrap and let rise until at least increased in size by 50

percent, 30 minutes to 1 hour.

While the bread is proofing, preheat the oven to 375F.

Whisk together the egg and brush it all over the top of the dough. Bake the brioche until it's deeply golden and domed on top, 25 to 30 minutes.

Remove the pan from the oven, slide the brioche out of the pan, and let it cool completely before slicing.