Step 1

Preheat the grill to medium heat.

Step 2

In a large skillet over medium heat, heat the olive oil.

Step 3

Add the onions and 1/4 teaspoon of the salt to the heated olive oil and cook until they are softened, about 6-8 minutes.

Step 4

Reduce the heat to medium-low and cook, stirring occasionally, until the onions are a deep golden-brown, about 30-40 minutes.

Step 5

Add the rosemary to the cooked onions and stir to combine.

Step 6

Transfer the cooked onions from the stovetop to a wire rack.

Step 7

Drizzle both sides of the chicken breasts with the lemon juice and sprinkle with the black pepper and the remaining salt.

Step 8

Add the chicken breasts to the grill and cook until they reach an internal temperature of 165 degrees F, about 5-7 minutes per side.

Step 9

Use a sharp knife to cut the chicken breasts into thin strips.

Step 10

In a small bowl, add the mayonnaise and the roasted red peppers and stir to combine.

Step 11

Evenly spread 1/2 of the mayonnaise mixture over 4 slices of the sourdough bread.

Step 12

Top the mayonnaise layer with 1 slice of the Swiss cheese, the chicken breast strips, the cooked onions, and 2 more slices of the Swiss cheese.

Step 13

Spread the remaining mayonnaise mixture over the remaining slices of sourdough bread and place the slices on the cheese-topped sandwiches.

Step 14

Spread the butter on the exteriors of both sides of the sandwiches.

Step 15

On the grill over medium heat, add the sandwiches and cook until they are golden-brown and the cheese has melted, about 2-3 minutes on each side.

Step 16

Plate and serve.