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Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess.
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To make the dough: In a stand mixer fitted with the dough hook attachment, mix together all the ingredients (except the butter) to make a smooth, stretchy dough that cleans the sides of the bowl; this should take 7 to 10 minutes at medium speed.
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Once the dough has developed some strength and elasticity, add one piece of the butter and mix at medium speed until it’s fully incorporated into the dough, 1 to 2 minutes. Scrape the bowl if necessary, then continue adding the butter one piece at a time, mixing to incorporate fully before adding the next piece. This dough takes longer than most to develop so be prepared to let the dough knead for another 5 to 10 minutes, or until it's formed a soft, smooth ball. We don't recommend trying to knead this dough by hand.
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Form the dough into a ball (it'll be very soft), place it in a greased bowl, cover the bowl, and it let rise for 1 hour. Then refrigerate the dough for several hours, or overnight. This will slow the fermentation and chill the butter, making the dough easier to shape.
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Divide the chilled dough into 12 pieces to make mini-brioche (brioche buns); leave it whole for one large round brioche; or divide it in half for two 8 1/2" x 4 1/2" loaves.
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Place the dough into the greased pan(s) of your choice, cover lightly, and let rise for 2 1/2 to 3 hours, until it's doubled and looks very puffy. If you're making two loaves, it's fun to make simple three-strand braids, and set them in the loaf pans.
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To bake a large, round brioche: Place the pan into a preheated 400°F oven. After 10 minutes, reduce the oven heat to 350°F and bake for about 30 to 35 minutes more. Check the brioche after 15 minutes; tent with aluminum foil if it appears to be browning too quickly. Brioche should be a deep brown when done, should sound hollow when tapped, and will read 190°F at the center using an instant-read thermometer. (It's easy to underbake since it browns so quickly!) Remove the brioche from the oven, and after 10 minutes remove it from the pan to cool completely on a rack.
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To bake the mini brioches: Place the pan(s) onto a baking sheet for easy handling, as shown; then nestle that baking sheet into another baking sheet, to protect the buns' bottoms from over-browning. Bake the buns in a preheated 375°F oven for 25 to 30 minutes (tent after 10 minutes if they're browning too quickly). Remove them from the oven, let stand for 5 minutes, then turn them out onto a rack.
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To bake the loaves: Allow the loaves to rise until they've nearly reached the rim of the pan, about 3 hours. Set the loaf pans on a baking sheet, for easy handling. Bake in a preheated 350°F oven for 30 to 35 minutes, tenting with foil after 15 to 20 minutes. Bake until the internal temperature reaches at least 190°F in the center, and up to 205°F for a crustier loaf.
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To make the glaze: Combine the confectioners' sugar, vanilla, and salt. Add enough cream to make a pourable consistency. Drizzle the glaze over the baked loaves, if desired.