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Toast the cumin, coriander, and caraway in a dry cast-iron skillet over medium heat until fragrant, about 3 minutes. Stir in the olive oil, garlic, onion, and salt. Cook until the garlic begins to brown, about 2 minutes.
Stir in the Urfa biber and tomato paste and cook for 1 minute. Add the red wine vinegar and cook 2 minutes more.
Transfer the mixture to a food processor fitted with a standard S-blade and process until smooth, scraping down the sides as needed. Store in a tightly sealed glass container in the refrigerator for up to 3 months.
Trim the ends off the lemons. Slice each lemon into 8 wedges, removing any seeds as you go. Reserve as much of the juice as possible.
Layer the lemon wedges in a wide-mouthed, 16-ounce canning jar, covering each layer with salt. Pack the jar as tightly as possible, pressing down to release the lemons' juice as you go and leaving about 1/4 inch of headspace in the jar.
Cover the wedges with the reserved lemon juice from the cutting board and the ends. If your lemons do not release a significant amount of juice, top off the jar with the juice of another lemon.
Stash in the refrigerator for 4 days, then flip the jar over and age another 4 days before sampling. The peel should be nice and soft. Rinse before using. Expect peak flavor and texture after about a month. As long as they're kept refrigerated, preserved lemons should keep indefinitely.
Position an oven rack 5 inches from the top of oven and set the broiler to high.
Cut the peppers in half lengthwise, discard the seeds, and place cut-side down on a sheet pan. Broil until the skins are completely charred, about 10 minutes, rotating the pan halfway through. (Or char the peppers over gas burners, turning often.)
Transfer the peppers to a large bowl, cover tightly with plastic wrap and allow to steam for 15 minutes, then rub the skins off under running water. Drain and roughly chop into half-inch squares.
Using the large holes of a box grater, grate the tomatoes into a large bowl and stash near the cooktop.
Place an 11-inch high-sided saute pan over medium heat for 2 minutes. Add the oil and heat until shimmering, or until it reaches 335 to 350ºF. Stir in the garlic and salt and cook until the garlic begins to brown, about 1 minute.
Add the harissa, brown sugar, and preserved lemons, and cook, stirring vigorously, for 30 seconds. Follow with the peppers and tomatoes. Reduce heat to medium-low and simmer, stirring every 5 minutes, until the sauce reduces and is slightly thickened, about 20 minutes.
Remove the eggs from the refrigerator and crack each egg into its own small bowl or teacup. Place a kitchen towel on the underside of the lid to the skillet and tuck into place under the handle.
Reduce the heat under the sauce as low as it will go. (If you have a simmer burner, now is the time to use it.) Create 6 evenly spaced divots in a ring in the tomato mixture with the back of a large spoon, making them deep enough to hold the eggs without pushing through to the bottom of the pan. Try to make the divots as close to the center of the pan as possible. Add the eggs to the divots, then use a soup spoon to gently push sauce over the whites of the eggs, leaving the yolks exposed. Cover with the towel-wrapped lid and cook for 7 minutes, rotating the pan 180 degrees every 2 minutes.
Turn off the heat and let the eggs finish cooking, still covered, in the residual heat until the whites are firm and the yolks are still runny, about 5 minutes more. Garnish with parsley and serve immediately, straight from the skillet, with warm bread.