Prepare the caramelized onions: heat 1 tbsp olive oil in a medium skillet over medium heat. Add the thinly sliced red onion, a pinch of salt and 1 tsp sugar. Cook, stirring occasionally, for 10–12 minutes until the onions are deeply golden and very soft. (Why: Moderate heat and a little sugar speed and encourage the Maillard reaction and caramelization without burning—the goal is sweet, jammy onions that will contrast the salty cheese.)

Make garlic-thyme butter: in a small bowl, mash together 4 tbsp unsalted butter, minced garlic and thyme until smooth. Season lightly with salt and pepper. (Why: Softened butter holds aromatics so it spreads evenly and helps create a crisp, flavored crust when the sandwich is grilled.)

Assemble the sandwiches: Lay out 4 slices of sourdough. Spread about 1 tsp Dijon mustard on the inside of two slices (this gives a subtle acidic counterpoint to the richness). On each mustard-coated slice add half the cheddar and half the Gruyère (layering two cheeses gives nutty depth and excellent melt). Top with half the caramelized onions, 3–4 tomato slices, a pinch of salt and pepper, then a handful of arugula. Close with the remaining bread slices. (Why: Layering cheese directly on the bread helps it melt into the crumb and bond the sandwich; the arugula goes inside rather than on top to keep it from wilting too much.)

Butter the outsides: spread the garlic-thyme butter evenly on the outside faces of both sandwiches (about 2 tbsp butter per sandwich, 1 tbsp per side). Press gently. (Why: The flavored butter promotes even browning, a crunchy exterior and adds aromatic complexity.)

Grill the sandwiches: heat a heavy skillet or cast-iron pan over medium-low to medium heat. Place sandwiches in the pan and cook 4–5 minutes per side, pressing lightly with a spatula, until golden brown and cheese is melted. If the bread browns too fast before the cheese melts, reduce the heat and cover the pan for 1–2 minutes to trap heat and help the cheese melt. (Why: Moderate heat gives time for the interior to warm and cheese to melt without burning the buttered crust; covering the pan creates steam to finish melting.)

Rest and slice: transfer sandwiches to a cutting board and let rest 2 minutes (this keeps cheese from running out too much when you cut). Slice diagonally and serve immediately. (Why: A short rest lets the molten cheese settle so each slice holds together and the flavors marry slightly.)