Bring softened butter to room temperature if needed. Spread about half of the softened butter (approximately 1/2 tbs) evenly on one side of each bread slice. The buttered sides will be the exterior of the sandwich that contacts the pan.

Assemble cheese

On the unbuttered side of one bread slice, spread 1 tsp Dijon mustard evenly. Sprinkle the shredded cheddar (0.75 cup) and shredded mozzarella (0.25 cup) evenly over the mustard. Season the cheese lightly with 1/8 tsp salt and a pinch (about 1/16 tsp) of freshly ground black pepper. Top with the second slice of bread, buttered side facing out.

Pan

Heat pan

Place a nonstick or cast-iron skillet over medium-low to medium heat. Add 1 tsp unsalted butter to the pan and let it melt and coat the surface. Heat until the butter foams but does not brown.

Grill sandwich

Place the assembled sandwich in the skillet, buttered side down. Press gently with a spatula to ensure even contact. Cook for 3 to 4 minutes until the bottom is golden brown. Reduce heat slightly if the bread is browning too quickly before the cheese melts.

Flip and finish

Carefully flip the sandwich using a spatula. Press down gently and cook the second side for another 3 to 4 minutes until golden brown and the cheese is fully melted. If the cheese is slow to melt, cover the skillet with a lid for 1 to 2 minutes to trap heat, which will help melt the cheese without burning the bread.

Rest and slice

Transfer the sandwich to a cutting board and let it rest for 1 minute to avoid the fillings spilling out. Slice diagonally or down the middle and serve immediately.

Optional (for serving)

Serve the grilled cheese with warmed tomato soup (heat separately to desired temperature) and a side of pickles or salad greens if desired.