Preheat the broiler. Bring a large skillet of water to a boil.
Thinly slice the cucumbers on a mandoline. Combine the cucumbers, parsley and dill in a medium bowl. Set aside.
Put the bread slices on a baking sheet. Brush both sides of the bread with the olive oil and broil until toasted and golden brown, 1 to 2 minutes per side.
Top the toast with a generous amount of the za'atar, cut each slice in half on a cutting board and set the slices back on the baking sheet.
Put the yogurt in a medium bowl. Stir in the juice and zest of 1 of the lemons. Season with kosher salt and pepper. Set aside.
Crack an egg into a fine-mesh strainer, straining out the thin egg white. Reduce the heat under the skillet until the water just simmers, and place the egg directly into the water. Repeat with 3 more eggs. Cook until the whites are set and the yolks still feel a bit runny when you delicately push down on them with your index finger, 2 to 3 minutes.
Remove the eggs from the water with a slotted spoon or spider and put them on a paper-towel lined plate to drain any excess water. Repeat with the remaining 4 eggs.
Peel the zest off the remaining second lemon with a vegetable peeler. Thinly slice the zest into fine julienne and set aside for garnish.
Add 1 tablespoon olive oil and the juice of 1/2 of the peeled lemon to the medium bowl with the cucumbers and herbs. (Reserve the remaining lemon half for another use.) Season with kosher salt and pepper and toss.
Dollop and then swoop a generous amount of the labneh mixture into a shallow bowl or deep plate. (To achieve a gorgeous swoop, use the back of a spoon.) Top with 2 poached eggs, drizzle with chili oil, if using, and place a nice handful of cucumber-and-herb salad next to the eggs. Rest 2 toast halves on the side of the bowl. Garnish with flaky sea salt, freshly cracked pepper and the julienned lemon zest. Repeat with the remaining ingredients.
Za'atar is a spice blend available at Middle Eastern markets.
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