Place the broccoli and broth (or water) in a small pot over high heat. Boil until broccoli is cooked through. Transfer to a cutting board and slice the stems off the florets and then finely chop the florets. Set aside.

Place a small skillet over medium heat and add in half of the butter. Once butter melts, add a slice of bread and top with broccoli, spreading it evenly out over the piece of bread. Top with cheese slices and other piece of bread. Press firmly down and let cook for 2-3 minutes. Flip, compress again with a spatula and let cook for 3-5 minutes or until cheese is fully melted.

Transfer the sandwich to a cutting board and slice according to your kids' preferences. Serve.