Remove the rind and any pale flesh from the watermelon. Remove the seeds and place 1 lb of the reddest watermelon into a blender. Process until smooth then pour into a strainer. You need 1 2/3 cups of watermelon juice.
Wash the strawberries, remove the hulls and cut in quarters. Add to watermelon juice in the blender. Add the wine, and a pinch each of sugar, salt and pepper, and process until very smooth. Refrigerate at least 3 hours (or, if you need it at once, add 2-3 ice cubes and process until smooth).
Line a baking sheet with parchment. Place goat cheeses on prepared sheet. Bake in a preheated 400F oven for 3-5 minutes, until warm and soft, but still white (do not allow to brown).
Pour the soup into shallow soup bowls, then add a cheese disc to each bowl. Top with a tiny drizzle of high-quality extra-virgin olive oil, a pinch of black pepper, and mint leaves.