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1. Line a large rimmed baking sheet with parchment paper. Using a sharp knife, carefully remove the salmon skin from the fillet and reserve. Thinly slice the salmon across the grain a scant 1/4 inch thick.
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2. In a medium bowl, whisk the sugar with the Aleppo pepper and the 1/2 cup of kosher salt. Spread half of the sugar mixture on the prepared baking sheet and arrange the salmon slices on top. Sprinkle the remaining sugar mixture evenly over the salmon and refrigerate until the fish is slightly firm, about 30 minutes.
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3. Meanwhile, preheat the oven to 325Au00b0. Spread the salmon skin on a parchment paper-lined baking sheet, silvery side up, and sprinkle with togarashi. Cover with a sheet of parchment paper and another baking sheet to keep the skin flat. Bake for 30 minutes, until crisp. Remove the top baking sheet and parchment paper and let the salmon crackling cool completely, then break into bite-size pieces.
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4. Fill a large bowl with cold water. Rinse the cured salmon in the water, rubbing off the sugar mixture. Transfer the salmon to paper towels to drain; pat thoroughly dry.
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5. In a small bowl, whisk the yogurt with the mint and season with salt. Arrange the salmon slices on plates or on a platter and garnish with the cracklings and roe. Serve with the mint yogurt.
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MAKE AHEAD The salmon can be prepared through Step 4 and refrigerated overnight.
