In a large bowl, combine water and yeast. Let stand for 5 to 10 minutes or until softened.
Meanwhile, in a saucepan, heat and stir the milk, shortening, sugar, and salt until the shortening is almost melted (120 degree F to 130 degree F).
Transfer the milk mixture to a large bowl. Stir in 2 cups of the flour; beat well by hand. Add the yeast mixture; stir until smooth.
Add mashed potatoes to the mixture along with the eggs and tarragon; beat until well mixed. Stir in as much of the remaining flour as you can.
Turn out onto a lightly floured surface. Knead in enough of the remaining flour to make a stiff dough that's smooth and elastic (8 to 10 minutes total). Place in a greased bowl; turn once to grease the surface. Cover and let rise in a warm place until doubled (about 1-1/2 hours).
Grease two 13x9x2-baking pans or one 13x9x2-inch baking pan and two 8x8x2-inch baking pans. Set aside.
Punch dough down. Shape into about forty 1-1/2-inch balls. Arrange in prepared pans. Cover and let rise until nearly doubled (30 to 40 minutes).
Bake in a 375 degree F oven for 15 to 18 minutes or until the rolls are golden. Remove from the oven; brush the tops with butter. Makes 40 rolls.