In the bowl of a stand mixer, whisk together the flour, sugar, yeast, and salt.

Add the butter, mashed potatoes, milk, and egg. Using the paddle attachment, mix on low speed for 30 seconds to combine the ingredients, then increase the speed to medium-high and mix for an additional 3-5 minutes, until the dough is pulling away from the sides of the mixer. Use a spatula to scrape down the sides of the bowl if the dough is sticking.

Switch to the dough hook and kneed on medium speed for 6-8 minutes. The dough will be tacky and soft. Form into a ball and place in a large, lightly greased bowl (you can use the stand mixer bowl if you clean it).

Cover with plastic wrap or a damp kitchen towel. Let rise for 2 hours (1 hour if using fast-acting instant yeast) or overnight in the refrigerator. The dough should double in size.

Punch down the dough. On a lightly floured surface, form the dough into a log and place in a lightly greased 8x4-inch loaf pan. Cover with lightly greased plastic wrap and let rise until the dough is about 1-inch above the top of the pan, 1-3 hours (it will take longer if the dough is chilled).

Preheat the oven to 350 degrees F and place an oven rack on the center shelf. Remove and discard the plastic wrap. Bake for 40-45 minutes, until the bread is a deep golden brown on top. If the top starts browning too quickly, you can tent it with foil after 20-25 minutes.

Let the bread cool for 10 minutes in the pan, then transfer to a wire cooling rack. Cool to room temperature before serving.