Heat 1 teaspoon of olive oil in a medium nonstick skillet over medium-high heat. Add tomatoes and cook for 3 to 4 minutes or until they begin to blister. Remove from heat and let stand 5 minutes.
In a large bowl, mix together vinegar, remaining olive oil, kosher salt, cold water and salt and pepper to taste.
Pulse tomatoes in a blender until mostly smooth and transfer to a large bowl. Pulse one of the cucumbers, jalapeño, garlic, and cherries in a blender until finely chopped. Combine the pulsated tomatoes, vinegar mixture, and cherry mixture in a large bowl. Cover and let stand at room temperature for at least one hour while the flavors combine.
Finish with slivered cucumbers, roughly chopped cherries, and a drizzle of olive oil. Salt and pepper to taste.
Cut the cucumber into ½ inch pieces and place in a colander. Sprinkle with salt and let sit for 10 minutes.
While the cucumber is draining, peel the avocado and chop into ½ inch pieces. In a large salad bowl, add avocado and coat with lime juice and roughly chopped apple mint. Add the cucumbers and salt and pepper to taste.