In a large skillet over medium-high heat, heat oil. Add sausage and cook until browned, 3 to 5 minutes, breaking up the meat as it cooks. Drain fat, remove from skillet and clean skillet.
Whisk together eggs and half-and-half; add egg mixture to skillet. Gently stir eggs using a wooden spoon or spatula and cook until they just begin to set. Season with salt and pepper and stir in chives. Remove eggs from skillet and wipe skillet clean.
Spread your choice of Cian’s mustard onto two bread slices, followed by the sausage, eggs, and cheese. Top with remaining two slices bread.
Return skillet to medium-low heat and add butter. When the butter begins to foam, add sandwiches. Cover the pan with a large lid (to help melt the cheese) and cook until bread is golden brown, about 2 minutes. Flip and cook until the cheese has melted and both sides of bread are golden, about 2 minutes more. Slice and serve immediately.
NOTE: Mayo can be used in place of butter to grill the sandwich. If using mayo, prior to step 3 spread mayo onto the outside of the bread slices. Follow with the remaining directions, omitting the use of butter in the skillet.