Cut the 10 strawberries into cubes (this should be about 1 cup). By the way, did you know that strawberries are the only fruit with their seeds on the outside?
Add the strawberries, tablespoon of white balsamic vinegar, 2 tablespoons of sugar to a small frying pan and warm it over a medium high heat. Stirring frequently, cook for about 5 minutes until the strawberries just begin to break down and lose their structure. Remove them from the heat to cool. Want to know the difference between balsamic vinegar and white balsamic vinegar? Balsamic vinegar is made from sweet, white Trebbiano grape pressings (called "must") that are simmered for hours until they caramelize and become thick. That syrup is then aged for character. Authentic balsamic is aged for no less than 12 years. White balsamic is cooked at high pressure to prevent caramelization, giving the vinegar a golden color and gentler flavor and is only aged for one year.
Warm 2 tablespoons of olive oil over a medium-low heat then add in the 3 sprigs of oregano and the pinch of salt. Turn the heat up to medium-high and add in the blueberries, pan frying them until the pan is covered with specks of indigo. This means the berries are cooked.
Place four slices of bread on a parchment-lined baking sheet. Cover two slices with the strawberry compote. Cover the other slices with blueberries. You don't want the olive oil or oregano, just the berries.
Divide the sharp white cheddar cheese amongst the four slices, stealing a few bits for yourself here and there. I mean, c'mon...you do all the work, you deserve some cheese!
Turn the broiler on high and then place the baking sheet in there. Be careful, you don't want to burn them. Cook it just long enough for the cheese to melt, about a minute.
While that is taking place, melt the tablespoon of butter in a cast iron skillet over a medium-low heat.
When the cheese has melted, remove the baking sheet and carefully build the sandwiches, placing a strawberry-covered slice on top of the blueberry one. Turn up the heat on the cast iron skillet to medium-high and then transfer the sandwiches to the pan. Fry until golden brown on each side.