Heat oil in a large skillet over medium heat. Add onion and cook, stirring frequently, until golden brown, 5 to 7 minutes. Add vinegar and cook until almost evaporated, about 1 minute. Add garlic and cook, stirring, until fragrant, about 30 seconds. Add kale, water and salt (the pan will be full). Stir, cover and cook, stirring occasionally, until the kale is wilted and the water has evaporated, about 5 minutes. Remove from heat.
To prepare panini: Preheat panini maker to high. Coat one side of each slice of bread with cooking spray. With the sprayed side down, spread the kale mixture on 4 slices of bread (about 1/2 cup per sandwich). Top each with 1/4 cup cheese and 2 slices tomato. Top with the remaining bread, sprayed-side up. Press in the panini maker until crispy, 3 to 5 minutes.