Put potatoes, water, and ¼ teaspoon of salt in a medium saucepan and bring to a simmer over medium heat, then turn heat down and cook potatoes until fork-tender, about 15 minutes. Transfer potatoes and their liquid to a bowl and let cool.

Add remaining 1 teaspoon salt, milk, yeast, and sugar to potatoes and stir to combine. Put potato mixture and eggs in a blender. Turn blender on low, gradually increase speed to medium-high, and purée until mixture is smooth, about 30 seconds.

Transfer potato mixture to bowl of a stand mixer fitted with paddle attachment (or use a hand mixer), add flour, and mix on low until all of potato mixture has been absorbed by flour. Increase speed to medium-high and beat until dough appears smooth and elastic, about 30 seconds. Use a rubber spatula to scrape dough into a clean bowl. Cover with plastic wrap and let it rise for 2 ½ hours, until not quite doubled.

Grease a 13- by 4- by 4-inch pain-de-mie or Pullman loaf pan and lid. Use a rubber spatula to stir dough together and then transfer it to prepared loaf pan. Cover loosely with plastic wrap and let rise until it reaches top of pan, about 2 hours. Meanwhile, 1 hour before baking, preheat oven to 400°F (200°C).

Remove plastic wrap from pan and slide lid into place. Bake bread for 45 minutes. Turn heat down to 350°F (180°C) and bake for 30 minutes more.

Turn heat down to 300°F (150°C). Remove top of pan and bake bread uncovered for 20 minutes.

Remove bread from oven and invert it onto a rack set over a baking sheet. Lift off pan and bake bread for 10 more minutes. The internal temperature should be 210° to 212°F (99° to 100°C). Turn off oven, open door, and let bread cool on rack for 30 minutes.

Remove bread from oven, transfer to a rack, and let cool to room temperature completely before cutting.