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In a small saucepan over low-medium heat, cook together 1/4 cup milk with 2 tablespoons of flour, whisking constantly, until a thick paste forms. Remove and cool completely. In a microwave safe bowl, heat together the remaining 3/4 cup whole milk with the brown sugar until it is about 110 degrees, or warm to the touch. Stir in the yeast, and set aside until puffy and bloomed.
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In the bowl of your stand mixer, mix together the egg and softened butter, and then stir in the bloomed yeast mixture. Then add in 3 cups flour and the salt, and stir until a shaggy dough forms. Add in the cooked flour paste and begin to knead the dough.
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Knead the dough with the dough hook for about 10 minutes or until the dough is soft and smooth. It should be still a little wet, and may stick to the bottom of your bowl slightly while kneading. Avoid adding too much flour, but you can add a bit more if it’s just globbing entirely on the bottom of the bowl.
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Scrape the dough into a ball, and cover the bowl with plastic wrap. Let the dough rise for about 30 minutes, or until it has almost doubled in size.