In a bowl (if using microwave) or small pot on the stove combine broth, smashed garlic cloves, 1/2 of one lemon, 1/2 Tablespoon Kosher salt. Cover tightly with lid or saran wrap and bring to a simmer for 5 minutes.
Add the halved potatoes, recover tightly and simmer until potatoes are fork tender (about 5-8 minutes depending on the size of your potatoes). Drain and discard lemon half and garlic.
In a bowl add the butter then the potatoes, carefully swirl the potatoes around in the bowl until butter is melted and has coated all the potatoes.
Sprinkle the garlic salt and ground black pepper (to taste). With the last half of the lemon squeeze juice (about 1 tablespoon) over the top of the potatoes and once again carefully "swirl" potatoes around in the bowl to coat.
On a preheated greased grill or oven (on greased baking sheet for oven) at 350 degrees F place potatoes cut side down for 10 minutes, turn and cook for another 5 minutes. (watch the color of the potatoes on the grill as they may be cooked faster than the oven).
Remove from heat and let potatoes rest 5 minutes before eating.