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1
Lightly spray a large bowl (preferably glass or metal) with nonstick spray and set aside. Combine the salt, sugar and pepper in a small bowl.
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2
Combine 4 of the eggs, milk and the yeast in the bowl of a stand mixer fitted with the dough hook and mix on medium-low until well combined. Add in the flour all at once and mix on low until a shaggy mess forms, about 2 minutes. Crank the speed to medium-high and beat for 1 1/2 minutes. As the mixer runs, stream in the salt mixture and mix for an additional 1 1/2 minutes. The dough should be taut and only slightly tacky. If it seems sticky, add in an additional ounce of flour. Lower the speed to medium-low and mix for an additional 2 minutes.
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3
With the mixer running, add in the butter and mix until it is completely incorporated, another 2 minutes or so. Scrape the dough into the prepared bowl, lightly spray a sheet of plastic wrap with nonstick spray and cover the bowl tightly.
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4
Allow the dough to rise for 1 hour. Remove the plastic wrap and punch the dough down completely. Return the plastic wrap and allow the dough to rise again for another hour. Punch the dough down one last time before covering and chilling overnight, or at least 8 hours.
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5
Line a large sheet pan (or 2 small) with parchment paper and lightly spray with nonstick spray. Pull the dough from the fridge and dump out onto a lightly floured work surface. Working quickly, pat the dough out into a rough rectangle. Portion the dough into 10 equal portions (approx. 3 oz each), using a bench scraper or sharp knife. Tuck the corners of each portion of dough under itself, pinching to form a ball. Roll the ball out on the work surface in a cupped hand to tighten the ball. Arrange the balls evenly on the prepared sheet pan(s) with at least 2 inches space in between them. Cover loosely with a clean towel or lightly sprayed piece of plastic wrap and store in a warm spot until they've doubled in size, 1 1/2 - 2 hours, depending on the temperature of your kitchen.
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6
Preheat the oven to 350? and arrange a rack to the center position. Combine the remaining egg with 1 T. water in a small cup and beat with a form until well combined. Gently remove the towel/plastic from the proofed dough and, using a soft pastry brush, glaze each bun with the egg wash. If using sprinkle each bun with the flaked sea salt. Bake the buns for 10 - 12 minutes, rotating the pan halfway through, or until the surface of the brioche is gorgeously golden-brown. Allow the buns to cool completely before serving or storing.
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7
Portion the ground lamb into 6 equal (approx. 6 oz. each) mounds. Gently form each mound into a patty, slightly indented in the middle. I am always careful not to pack to meat too tightly, though. Season each side generously with salt and pepper and allow to rest while you prepare to cook.
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8
Heat 1/4-inch oil in a large cast iron or stainless steel skillet over medium heat until hot, about 5 minutes. While that's heating whisk together the yogurt, mustard, garlic, mint, cumin and cinnamon with a pinch of salt and a few good grinds of black pepper. Set aside with the lettuce, tomato, onion and feta.
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9
When the oil in the pan is hot, carefully lay a few burgers (as many as will fit and still have 1-inch space around each patty) in and take a step back. Do not move, press, poke or even touch the meat. There is nothing you can do right now, to improve the process. Allow the patty to get its crisp crust, undisturbed, for about 4 minutes. Using a thin spatula, flip each burger over and cook for another 3 - 4 minutes (for medium). Remove the burgers to a plate, tent with foil and allow to rest for 5 minutes before building.
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10
How I stack a burger: Slice 6 brioche buns in half, toast if desired, and smear the bottoms with a generous dollop of the yogurt sauce. Lay a large piece of lettuce on top of the sauce, folding it over if necessary to allow it to fit. Place on top of the lettuce a few rings of red onion, then a warm burger, then a few slices of tomato. The finishing touch is a few shards (or generous 5-fingered pinch) of salty feta and the crowning bun. Enjoy immediately.
