Preheat the oven to 425 degrees . Line a baking sheet with parchment paper.
In a large skillet, cook the sugar, water and butter over high heat until bubbling and a light amber caramel forms, 6 to 8 minutes. Add the apples and cook over medium-low heat, turning occasionally, until softened, about 12 minutes. Transfer the apples to a plate and let cool. Let the caramel cool, them pour in into a small bowl and set aside.
Spread the bread slices on a clean, dry work surface and scatter a rounded tablespoon of the shredded cheddar cheese evenly over each slice. Press and roll the cheese into the bread, using a rolling pin, until the cheese and bread are flattened into thin squares. Turn the bread slices cheese side down.
For each turnover, lightly wet the edges of 2 adjacent sides of the bread with water, using your finger or a pastry brush. Place 1 cooled apple quarter in the center of each slice and spoon a little caramel on top. Fold the slice diagonally in half, forming a triangle in such a way that the dampened edges of the bread meet the dry edges, and using your fingers, press the edges together to seal. Transfer the turnovers to the prepared baking sheet and bake until golden and the cheese is crusty, about 20 minutes. Let cool before serving with a drizzle of the reserved caramel sauce.