Step 1

Preheat oven to 400°. Heat 2 Tbsp. butter in a small saucepan over high heat. When butter begins to foam, add shallots and thyme; season with salt and pepper. Cook, stirring continuously with a wooden spoon or spatula, until shallots begin to soften and caramelize, 4–5 minutes; remove from heat and set aside.

Step 2

Heat a large heavy-bottomed skillet over medium heat. Working in 2 batches, add 1 Tbsp. butter and swirl in pan to melt butter and coat bottom of pan. Add 2 slices of bread to pan and cook until golden brown and crisp on the bottom, 2–3 minutes. Transfer bread, toasted side down, to a rimmed baking sheet. Repeat with remaining butter and bread slices. Divide cheese evenly among bread slices; top cheese with reserved shallots.

Step 3

Place baking sheet in oven and bake until cheese is melted, 7–8 minutes.

Step 4

Combine arugula, apple slices, lemon juice, and oil in a large bowl; toss to coat and evenly distribute. Season salad to taste with salt and pepper.

Step 5

Press 2 pieces of bread together, melted cheese sides in; halve sandwich on a diagonal and place on a plate. Repeat with remaining bread slices. Divide salad between plates.