It is called "tzatziki" sauce. It is good, and I dislike cucumbers normally.
Here is one from cooks illustrated.
Greek-Style Lamb Pita Sandwiches with Tzatziki Sauce—Gyros
For a home-style lamb sandwich recipe that would give us the same flavors and textures as restaurant gyros, even if they didn't have the traditional appearance, we pan-fried ground lamb patties flavored with oregano, onion, and minced garlic. To make the patties juicier, we added a panade (a paste of fresh bread crumbs and milk), substituting pita crumbs for the traditional bread crumbs. For readers who can’t easily find ground lamb, we developed a ground beef version of our lamb sandwich recipe.
ingredients
For Tzatziki Sauce
1cup plain whole-milk yogurt
½ medium cucumber, peeled, seeded, and diced fine (about ½ cup)
⅜teaspoon table salt
1tablespoon juice from 1 lemon
1 small garlic clove, minced or pressed through a garlic press (about ½ teaspoon)
1tablespoon finely chopped fresh mint leaves or dill
For Lamb Patties
4 (8-inch) pita breads (see note above)
½ medium onion, chopped coarse (about ¾ cup)
4teaspoons fresh lemon juice
½teaspoon table salt
¼teaspoon ground black pepper
1tablespoon minced fresh oregano leaves
2 medium garlic cloves, minced or pressed through a garlic press (about 2 teaspoons)
1pound ground lamb
2teaspoons vegetable oil
Accompaniments
1 large tomato, sliced thin
2cups shredded iceberg lettuce
2ounces crumbled feta cheese (about ½ cup)
Instructions Serves 4
Since the yogurt and cucumbers in the Tzatziki Sauce need to drain for 30 minutes, start making the sauce before the patties. Although we prefer the richness of plain whole-milk yogurt, low-fat yogurt can be substituted. Greek yogurt can also be substituted, but use 1/2 cup and skip the step of draining.
While we didn’t like the flavor of dried mint, dried dill may be used in place of fresh, but reduce the amount to 1/2 teaspoon. For the patties, the test kitchen prefers the flavor of fresh oregano, but 1 teaspoon of dried can be substituted. The skillet may appear crowded when you begin cooking the patties, but they will shrink slightly as they cook.
If using pocketless pitas (see illustrations below for folding instructions), heat them in a single layer on a baking sheet in a 350-degree oven for 5 minutes.
Do not cut top quarters off pocketless pitas; instead, use a portion of a fifth pita to create crumbs in step 1.
When cooking the patties, use a splatter screen to keep the mess to a minimum. The patties can be prepared through step 1 and refrigerated for up to a day or frozen before cooking as directed in step 2 (frozen patties should be thawed in refrigerator prior to cooking). This recipe can be doubled.
1. For the Tzatziki Sauce: Line fine-mesh strainer set over deep container or bowl with 3 paper coffee filters or triple layer of paper towels. Spoon yogurt into lined strainer, cover, and refrigerate for 30 minutes.
2. Meanwhile, combine cucumber, 1/8 teaspoon salt, and lemon juice in colander set over bowl and let stand 30 minutes.
3. Discard drained liquid from yogurt. Combine thickened yogurt, drained cucumber, remaining 1/4 teaspoon salt, garlic, and mint in clean bowl.
4. For the Patties: Adjust oven rack to middle position and heat oven to 350 degrees. Cut top quarter off each pita bread. Tear quarters into 1-inch pieces. (You should have 3/4 cup pita pieces.) Stack pitas and tightly wrap with aluminum foil. Process onion, lemon juice, salt, pepper, oregano, garlic, and pita bread pieces in food processor until smooth paste forms, about 30 seconds. Transfer onion mixture to large bowl; add lamb and gently mix with hands until thoroughly combined. Divide mixture into 12 equal pieces and roll into balls. Gently flatten balls into round disks, about 1/2 inch thick and 2 1/2 inches in diameter.
5. Place foil-wrapped pitas directly on oven rack and heat for 10 minutes. Meanwhile, heat oil in 12-inch nonstick skillet over medium-high heat until just smoking. Add patties and cook until well browned and crust forms, 3 to 4 minutes. Flip patties, reduce heat to medium, and cook until well browned and crust forms on second side, about 5 minutes longer. (See below for tips on flipping patties.) Transfer patties to paper towel-lined plate.
6. Using soup spoon, spread 1/4 cup Tzatziki Sauce inside each pita. Divide patties evenly among pitas; top each sandwich with tomato slices, 1/2 cup shredded lettuce, and 2 tablespoons feta. Serve immediately.
(they have a variation with beef instead of lamb also, a little less lemon juice, double oregano, and an extra clove of garlic) Can also grill them.