In a 2 cup measuring cup, stir together the warm water, yeast and 1 tablespoon sugar. Let sit until risen to about the top of the measuring cup.
In a large bowl, stir together the sugar, salt, eggs and shortening. Stir in the yeast mixture.
Gradually add the flour. When it's getting hard to stir, dump it out on a counter and knead in enough flour to make it a dough you can work with (not super tacky). Knead for a total of 5-7 minutes.
Place the dough in a greased bowl; flip ball of dough so both sides get some oil on it.
Cover with a sheet of waxed paper or a light kitchen towel and let rise in a warm place for about an hour, or until risen to double its size.
Separate the dough into three equal balls. Roll each into a 12" circle and brush with melted butter.
Use a pizza cutter to cut each circle into 12 triangle pieces.
Roll the triangles from the outside of the circle to the center. Place the butterhorn on a parchment-lined pan, making sure the "tail" is tucked underneath.
Let the butterhorns rise for 45 minutes.
Then, bake at 350°F for 8-10 minutes or until a light golden brown. Brush with additional butter.