For the Potatoes: Combine the potatoes and the 2 cups pickle brine in a large bowl. Cover with plastic wrap and refrigerate overnight. When you’re ready to cook the potatoes, drain them well and pat dry with paper towels.
Heat 1/4 inch oil in a large skillet over medium heat until shimmering but not smoking.
Working in batches to avoid crowding the skillet, add the potatoes in a single layer and fry until brown and crisp on the outside and tender within, about 3 minutes per side.
For the Basic Tehina Sauce:
Break up the head of garlic with your hands, letting the unpeeled cloves fall into a blender. Add the lemon juice and 1/2 teaspoon of the salt. Blend on high for a few seconds until you have a coarse puree. Let the mixture stand for 10 minutes to let the garlic mellow.
Pour the mixture through a fine-mesh strainer set over a large mixing bowl, pressing the solids to extract as much liquid as possible. Discard the solids.Add the tehina to the strained lemon juice in the bowl, along with the cumin and 1 teaspoon of the salt.
Whisk the mixture together until smooth (or use a food processor), adding ice water, a few tablespoons at a time, to thin it out. The sauce will lighten in color as you whisk. When the tehina seizes up or tightens, keep adding ice water, bit by bit (about 1-1/2 cups in total), whisking energetically until you have a perfectly smooth, creamy, thick sauce.
Taste and add up to 1-1/2 teaspoons more salt and cumin, if you like. If you're not using the sauce immediately, whisk in a few tablespoons of ice water to loosen it before refrigerating. The tehina sauce will keep a week refrigerated, or it can be frozen for up to a month.
For the Harissa Paste: Combine Aleppo pepper, garlic, wine vinegar, cumin, coriander, caraway, and salt. Blend in a food processor to a coarse puree. Add cup canola oil and process for another few seconds. Stop short of making it perfectly smooth. Refrigerated, harissa paste will keep 2 weeks.
To Make the Harissa Tehina: Whisk together 1 cup of the tehina sauce and the 2 tablespoons pickle brine. Stir in 1/4 cup of the harissa — I like it when the sauce looks a bit broken and streaky. Serve the potatoes with the tehina sauce.