Lightly oil two 7x11x1 1/2-inch baking pans.,

Place the mashed potato, water, sugar, and yeast in a large bowl. Mix well with a wire whisk. Add 1 cup of the flour, and whisk until the mixture is quite smooth, about 2 minutes. (There may be some lumps from the potato, which is fine.) Add another 1/2 cup flour, and mix until smooth. Cover with a kitchen towel, set in a warm spot (68 to 70 degrees F), and let rise until doubled in bulk, about 1 hour.,

Add the eggs to this mixture, one at a time, mixing vigorously after each addition. Add the butter and salt, and mix well. Then gradually add 2 1/2 cups flour, mixing quite vigorously. When the dough becomes too awkward to mix in the bowl, turn it out onto a lightly floured surface, and knead it until it is smooth, 5 to 10 minutes. Add flour (no more than an additional cup) as you knead to keep the dough from sticking to the work surface and your hands. It will still be very soft-softer than bread dough.,

Heavily flour the dough and the work surface, and roll the dough out until it is 3/4 inch thick. Cut the dough into 2-inch squares and place them in the prepared pans, leaving about inch of space between them. Cover loosely with a damp kitchen towel and set aside to rise in a warm spot (68 to 70 degrees) until they have nearly doubled in bulk, 1 to 2 hours.,

Preheat the oven to 350 degrees F.,

Bake the rolls in the center of the oven until they are golden and puffed, about 12 minutes. Turn them out of the pans onto a wire cooling rack, let cool for 5 minutes, and then serve.