Blend the mashed squash with the 1/3 cup of water until it is smooth. Set Aside.
Combine the milk, sugar and yeast in mixing bowl of Kitchen Aide. Let stand for 8-10 minutes to dissolve yeast. While that is working, add the oil, lecithin, salt and butter flavoring just in the edge of the bowl.
Add 3 cups of the flour, the gluten flour and the blended squash, mixing on speed 2 for two minutes. (If mixing by hand I recommend using a hand mixer for 4-5 minutes.) Let this stand for 10 minutes.
Beat in the additional flour as needed to make a soft dough that does not stick to the sides of the bowl. Continue to beat for 2 minutes. (If mixing by hand, add flour, mixing and kneading until dough is formed. Continue to knead for 8-10 minutes.)
Cover and let rise until double, about 1 hour.
Punch down and form into dinner rolls and place in a 9" x 14" glass baking dish prepared with food release. Space rolls evenly with room to rise. Cover and let rise until double, about 30 minutes depending on temperature of your kitchen.
Bake at 375 degrees for approximately 20-25 minutes, until golden brown. Turn out onto a cooling rack.
Preferably this milk should be a rich milk. I use purchased almond milk that I enrich by blending with an additional 2 tablespoons of almond flour. It works well for this and also is delicious in many other applications.