Place blueberries, balsamic glaze, and sugar in a small saucepan over medium heat.
Simmer for 5 minutes until berries burst and mixture thickens into a jam-like consistency.
Spread butter on one side of each bread slice.
Flip slices over and spread goat cheese on the unbuttered sides.
Layer white cheddar and the balsamic blueberry mixture onto two of the slices and sprinkle with thyme.
Top with the remaining bread slices, buttered side facing out.
Heat a skillet over medium-low heat.
Grill sandwiches for 3 to 4 minutes per side until the bread is golden brown and the cheese is melted.