In the bowl of a stand mixer fitted with the paddle attachment, beat 1½ cups (188 grams) all-purpose 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 120–130°F (49–54°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 at low speed until well combined, about 1 minute.
With mixer on low speed, gradually add whole-wheat flour and remaining 1 cup all-purpose 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, 12–15 minutes; add up to remaining ¼ cup (31 grams) all-purpose flour, 1 tablespoon (8 grams) at a time, if dough is too sticky (*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 with clean kitchen towel and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 40–50 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 5–10 minutes.
Turn out dough onto a clean surface; divide into 12 portions (about 86 grams each). (Keep portions covered as you work.) On a clean work surface, shape each portion into a smooth dough ball. Using your hands, pat each ball into a 3-inch disk. Place 6 dough disks, evenly spaced, on each prepared sheet. Cover with clean kitchen towel and let rise in a warm, draft-free place (75°F/24°C) until puffed and tops are rounded, 25–30 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 dough rounds with za’atar and additional flaky sea salt (or kosher salt).
Bake, one batch at a time, until golden brown and an instant-read thermometer inserted in center registers 200°F (93°C), 13–15 minutes, rotating pan halfway through baking. Serve warm or at room temperature. Best served same day.