Roast jalapeño halves in the oven at 400 F until nicely charred, about 30 minutes. Remove from the oven and let cool.
Slice mostly through each chicken breast, but not all the way, to make a pocket. Stuff with roasted jalapeños, mozzarella cheese, and half of the scallions. If needed, secure the stuffed chicken breasts with toothpicks.
Heat a pan with olive oil over medium-low heat. Add the stuffed chicken breasts to the pan. Cover with a lid. Cook until the chicken is cooked through, about 10 minutes per side.
While the chicken is cooking, combine and mix the honey balsamic glaze ingredients in a saucepot. Bring to a boil and then reduce the heat to a simmer. Let it simmer until the glaze reduces to half its volume, stirring occasionally, about 20 minutes.
Pour the honey balsamic glaze over the chicken in the pan. Sprinkle with remaining scallions.