Toss the jalapeño halves on a baking sheet with the olive oil and season with salt and pepper. Roast until charred and softened, about 15 minutes. Set aside.
Heat a griddle or nonstick skillet over medium heat.
On each of 4 bottom slices of bread, layer about 2 tablespoons shredded cheese, then 4 jalapeño halves, then about 4 chips, then 1/2 cup of the chicken and then the remaining cheese. Top each with another slice of bread.
Spread both sides of the sandwiches with the butter. Cook the sandwiches, doming them with a metal bowl to help melt the middles, until the first side is golden brown, 2 to 3 minutes. Flip and repeat with the second side, until the bottoms are golden brown and the sandwiches are gooey in the middle, another 2 to 3 minutes. Cut in half and serve.