In a small saucepan, cook squash in water until tender. Drain and mash. Should make 1 cup.
In large mixing bowl, combine cooked squash, milk, butter, egg, brown sugar, and salt. Mix well.
Add yeast and 1 1/2 cups flour. Mix well. Add remaining flour until it makes a soft dough. Turn out onto a floured surface and knead until smooth and elastic, about 6-8 minutes. Place in greased bowl and allow to rise until double, about 1 hour.
Punch down dough. Divide into thirds. Roll each into an 18" rope. Place all three on a baking sheet lined with parchment paper or a silicone baking mat. Braid ropes together and pinch ends, folding ends under to seal. Cover and let rise until doubled, about 30-40 minutes.
In a small bowl whisk together egg and water to make egg wash. Brush over braid. Bake in preheated 350 degree oven for 20-25 minutes until golden brown. Cool on wire rack. Makes 1 loaf.
Notes
*Unless otherwise indicated, all butter used on this recipe site is salted.