Add eggs, salt, sugar, butter, mashed potatoes, yeast, water, and 3 cups of flour to a large bowl.
Mix until combined.
Continue adding flour until soft dough forms. You may not need to add all of the flour.
Place the dough in a lightly greased bowl, and cover the bowl with plastic wrap.
Let the dough rise until it's doubled, about 1-1½ hours.
Gently deflate the dough.
Add to heavily floured surface, and knead in additional flour to form a soft, workable dough.
Divide dough into 24 balls.
Place the rolls in a lightly greased 9x13-inch pan.
Cover the pan with lightly greased plastic wrap, and let them rise for about 1-1 ½ hours, or until they're puffy.
Towards the end of the rising time, preheat the oven to 350F.
Bake the rolls for 20-25 minutes, or until they're golden brown and feel set.
Brush with melted butter, if desired.
Notes
Sugar: The rolls are slightly sweet. If you would prefer a roll that's less sweet, decrease the sugar to 1/4 cup.
Butter: I use salted butter in the rolls. Feel free to use unsalted if you prefer.
Mashed potato: 1 medium baking potato will yield about 8 ounces of mashed potato. You can also reconstitute mashed potato flakes and use 1 cup of the prepared mashed potatoes.
Water: I like to use the leftover water from boiling the potato because it has extra starch in it. If you don't have potato water, use tap water that's about 80F.
All-purpose flour: Weighing the flour will give you the most accurate results. You can also sift or stir the flour to break it up, lightly spoon it into a measuring cup and avoid packing it in, and then level.