Adjust oven rack to middle position and heat oven to 200 degrees. When oven reaches 200 degrees, shut oven off. Grease large bowl with 1 tablespoon butter. Line 13 by 9-inch baking dish with foil, leaving overhang on all sides. Grease foil with 1 tablespoon butter.
Place milk, honey, shortening, and remaining 3 tablespoons butter in large measuring cup. Microwave on high power until milk is warm (110 degrees) and butter and shortening begin to melt, 1 to 2 minutes. Stir well.
Mix 4 1/2 cups flour, yeast, and salt in bowl of standing mixer fitted with dough hook. Turn mixer to low and slowly add milk mixture. After dough comes together, increase speed to medium, add 1 egg, and mix until dough is smooth, about 2 minutes. Add another 1/2 cup flour and knead until dough is shiny and smooth and comes away from sides of mixing bowl, 6 to 7 minutes (add up to 1/2 cup more flour if dough is too sticky). Turn dough onto unfloured surface and knead briefly to form smooth, cohesive ball. Transfer dough to buttered bowl and turn to coat. Cover bowl with plastic wrap and place in turned-off oven until dough has doubled in size, 50 to 60 minutes.
Punch down dough on floured surface and divide into 3 equal pieces. Roll each piece into thick cylinder and cut each cylinder into 5 equal pieces. Working one piece at a time (keep remaining pieces covered with plastic wrap) and following photos, form dough pieces into smooth, taut rounds and arrange in prepared baking dish. Lightly press on dough rounds so they just touch each other. Cover baking dish with plastic wrap and return to turned-off oven until dough rounds have doubled in size, 50 to 60 minutes.
Remove dish from oven and heat oven to 375 degrees. Unwrap baking dish and brush rolls with egg-water mixture. Bake until rolls are deep golden brown, 25 to 27 minutes. Remove from oven and cool on wire rack for 5 minutes. Using handles of foil sling, remove rolls from baking dish and cool on wire rack for 1 hour. Remove foil from rolls, return to rack, and cool completely, about 2 hours longer. Keeping rolls together, wrap tightly in plastic wrap, then in foil; freeze for up to 1 month.
To serve: Remove plastic wrap and foil (reserve foil) from rolls, wrap in reserved foil, and defrost at room temperature for 2 hours. Adjust oven rack to middle position and heat oven to 375 degrees. Bake foil-wrapped rolls on baking sheet until heated through, about 30 minutes. Serve.
Rolls without a Mixer: In step 3, mix 4 1/2 cups flour, yeast, and salt in large bowl. Make well in flour, then add milk mixture and egg to well. Stir with wooden spoon until dough becomes shaggy and difficult to stir. Turn dough out onto floured work surface and knead, adding remaining 1/2 cup flour (or more as needed), until dough forms smooth, cohesive ball, about 10 minutes. Proceed as directed.