Combine water, 1/3 cup white sugar, and yeast in the bowl of an electric mixer fitted with a paddle attachment. Add 2 cups flour and salt; beat until smooth, about 2 minutes. Add eggs and cooled shortening; beat until incorporated, about 1 minute. Gradually stir in remaining flour with a spoon until dough is firm. Let dough rest, about 20 minutes.

Turn out 1/3 of the dough on a floured work surface. Flatten out into a large rectangle. Sprinkle 1 cup white sugar and cinnamon over dough. Roll dough up tightly starting with the edge closest to you; pinch the seam to seal it at the end. Cut into 12 slices.

Heat butter in a small saucepan over low heat until melted, 2 to 3 minutes.

Pour melted butter into a 9x13-inch baking pan. Sprinkle 1 cup brown sugar evenly over butter. Arrange rolls cut-side up, slightly apart, in the pan. Cover loosely with plastic wrap and let rise until rolls have doubled, 1 1/2 to 2 hours.

Preheat oven to 350 degrees F (175 degrees C). Remove plastic wrap from baking pan.

Bake rolls in the preheated oven until golden brown, about 20 minutes. Invert pan onto heatproof serving plate. Leave pan over rolls for 1 to 2 minutes; remove pan.