Heat a heavy sauté pan over medium-high heat.
Add the unsalted butter to the pan. When it begins to foam, add the onions and sauté until tender and translucent.
Add the brown sugar, reduce heat to medium low, and cook, stirring frequently until caramelized.
Remove the onions from the heat and reserve.
Combine the mustard and apple butter, and reserve.
Heat a griddle over medium high heat.
Toast the slider rolls on the griddle, transfer to a plate, and reserve.
Heat the ham on the griddle, moving frequently until just warmed through and beginning to brown slightly.
To assemble: Spread a bit of the apple mustard over each of the bottoms of the rolls. Place two quarter slices of cheddar cheese on each. Divide the ham over the rolls, top with some of the onions, dress the tops of the rolls with some of the apple mustard, and top each sandwich.