Sprinkle the cubed cucumber with 1 teaspoon kosher salt, and allow to sit refrigerated for at least 45 minutes so some of the liquid can be extracted.
Meanwhile, combine tomatoes, garlic, onion, jalapeƱos and cilantro. Whisk together lime juice, olive oil, cumin and cayenne and stir into tomatoes. Marinate in the refrigerator for 30 minutes.
Preheat the oven to 350 degrees. Arrange cornbread cubes in a single layer on a baking sheet. Bake uncovered for 15- 20 minutes, or until dry and crisp.
Drain the cucumbers and toss with the tomatoes. Adjust seasonings and add salt to taste. (Wait to salt the salad until after adding the cucumbers because the cucumbers will be salty.) Top with crumbled Cotija cheese, and serve with cornbread croutons on the side, since the croutons will get soggy if they sit in the salad for too long.