Finely dice 1/3 of the peeled, deseeded cucumbers and store them in the refrigerator until ready to use.
Coarsely chop the remaining prepped cucumbers and set aside until ready to use.
Use the chiffonade method to cut all of the herbs. Set aside until ready to use.
Place half of the coarsely chopped cucumbers in a blender with ¼ cup water; blend well. With the blender running, add the remaining coarsely chopped cucumbers, shallots and garlic along with 1 ½ Tbsp. of the Kosher salt. Blend the mixture for at least 30-40 seconds, or until everything is well-blended.
Add the yogurt to a bowl large enough to hold double or triple its size. Add the puréed cucumber mixture. Using a whisk, blend well.
As you whisk, slowly drizzle in most of the olive oil and continue mixing until the oil is absorbed.
Take the diced cucumbers from the refrigerator and place in a separate bowl. To it, add the remaining 1 ½ Tbsp. salt, most of the herbs, lemon juice, and lime juice, if using. Toss well so everything is coated with the salt and juice.
Add the cucumber-herb mixture to the yogurt and puréed cucumber; mix well. Check for seasonings and adjust, if necessary.
To serve: Place ja’jik in individual bowls, add an ice cube, if desired. Garnish with some of the remaining herbs and a drizzle of olive oil.