- Heat oil in large, heavy skillet over medium-high heat. Add onions and sauté until softened, about 5 minutes.
- Stir in vinegar and sugar. Reduce heat to medium-low and cook until onions are soft and dark brown, about 20 minutes.
- Season to taste with salt and pepper.
(You can prepare this up to 2 hours ahead. If you do, hold at room temperature, then rewarm over low heat before using.)
- Preheat broiler. Brush large metal baking pan with oil.
- Place sliced sausages in pan. Broil until sausages are brown and crisp, about 5 minutes per side. Sausages may also be grilled over medium-high heat.
- To assemble sandwiches, place 4 slices of bread on work surface. Top each slice with one ounce of cheese.
- Place 3 sausage halves on each slice, and spread 1 teaspoon of mustard on top of sausages.
- Top sausages with caramelized onions and remaining cheese slices, dividing equally.
- Finish with remaining 4 bread slices. Spread half of butter over upper bread slices.
- Heat large nonstick skillet over medium-low heat, or electric griddle on medium.
- Very carefully ... invert sandwiches, buttered side down, onto skillet or griddle.
- Cook until bottoms are brown, about 3 minutes. Spread remaining butter over upper bread slices.
- Turn sandwiches, and cook until bottoms are brown, about 3 minutes longer.