Dough
Topping
FOR THE DOUGH Pulse flour, sugar, and yeast in food
processor until combined, about 5 pulses. With processor
running, slowly add ice water and process until dough is
just combined and no dry flour remains, about 10 seconds.
Let dough rest for 10 minutes.
Add oil and salt to dough and process until dough forms
satiny, sticky ball that clears sides of bowl, 30 to 60 seconds. Transfer dough to lightly floured counter and knead by hand to form smooth, round ball, about 30 seconds. Place dough seam side down in lightly greased large bowl or container, cover tightly with plastic wrap, and refrigerate for at least 24 hours or up to 3 days.
Press down on dough to deflate. Transfer dough to
lightly floured counter, divide into quarters, and cover
loosely with greased plastic. Working with 1 piece of dough
at a time (keep remaining pieces covered), form into rough
ball by stretching dough around your thumbs and pinching
edges together so that top is smooth. Space balls 3 inches
apart, cover loosely with greased plastic, and let rest for 1
hour.
FOR THE TOPPING Process pepper paste, tomato paste, garlic, paprika, allspice, and salt in clean, dry work bowl until well combined, about 20 seconds, scraping down sides of bowl as needed. Add bell pepper and onion and pulse until finely ground, about 10 pulses. Add lamb and parsley and pulse until well combined, about 8 pulses.
Adjust oven racks to upper-middle and lower-middle
positions and heat oven to 350 degrees. Grease 2 rimmed
baking sheets. Generously coat 1 dough ball with flour and
place on well-floured counter. Press and roll into 9-inch
round. Arrange round on prepared sheet, with edges fitted
snugly into 1 corner of sheet, and reshape as needed. (If
dough resists stretching, let it relax for 10 to 20 minutes
before trying to stretch it again.) Repeat with remaining
dough balls, arranging 2 rounds on each sheet in opposite
corners.
Using back of spoon, spread one-quarter of topping in
thin layer on surface of each dough round, leaving ¼-inch
border around edge.
Bake until edges of flatbreads are set but still pale, 10 to
12 minutes, switching and rotating sheets halfway through
baking. Remove flatbreads from oven and heat broiler.
Return 1 sheet to upper rack and broil until edges of
flatbreads are crisp and spotty brown and filling is set, 2 to
4 minutes. Transfer flatbreads to wire rack with spatula and
let cool for 5 minutes before serving. Repeat broiling with
remaining flatbreads.