In the bowl of a stand mixer fitted with the paddle attachment, beat 1½ cups (188 grams) flour, sugar, salt, and yeast at low speed until combined.

In a medium saucepan, combine ½ cup (120 grams) water, milk, and butter; cook over medium-low heat until butter is melted and an instant-read thermometer registers 120°F (49°C) to 130°F (54°C). Add warm milk mixture to flour mixture; beat at low speed for 1 minute, stopping to scrape sides of bowl. Add mashed potatoes and 1 egg (50 grams); beat until well combined, about 1 minute. With mixer on low speed, gradually add 3 cups (375 grams) flour, beating just until a shaggy dough forms.

Switch to the dough hook attachment. Beat at low speed until a soft, elastic, and slightly tacky dough forms, 8 to 10 minutes; add up to remaining ¼ cup (31 grams) flour, 1 tablespoon (8 grams) at a time, if dough is too sticky. (Dough should pass the windowpane test; see Note.) Turn out dough onto a lightly floured surface, and shape into a smooth round.

Lightly oil a large bowl. Place dough in bowl, turning to grease top. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 30 to 45 minutes.

Position oven rack in top third of oven. Preheat oven to 400°F (200°C). Line 2 baking sheets with parchment paper.

Punch down dough; cover and let stand for 5 to 10 minutes. Turn out dough onto a clean surface; divide into 12 portions (about 90 grams each). (Keep portions covered as you work.) On a clean surface, shape each portion into a smooth ball. Using your hands, pat each ball into a 3-inch disk. Place dough disks evenly spaced on prepared pans. Cover and let rise in a warm, draft-free place (75°F/24°C) until puffed and tops are rounded, 15 to 25 minutes.

In a small bowl, whisk remaining 1 egg (50 grams). Working with one batch at a time, brush tops and sides of dough with egg wash. Sprinkle with everything bagel seasoning.

Bake, one batch at a time, until golden brown and an instant-read thermometer inserted in center registers 200°F (93°C), 12 to 15 minutes, rotating pan halfway through baking. Serve warm or at room temperature. Best served same day.