Heat a foil-lined outdoor grill or an indoor grill pan over medium-high heat. Add the butter or EVOO and onions, reduce to medium and cook, stirring occasionally, until the onions soften, 5 minutes. Add the garlic and thyme and cook, stirring, 3 minutes. Add the wine and simmer until evaporated, about 5 minutes. Add the broth and cook until onions are very tender, 10 minutes. Season with salt and pepper.
Preheat the broiler. Form the beef into four 4-inch patties and season on both sides. Heat an outdoor grill or an indoor grill pan over medium-high heat, add the burgers and cook about 2 1/2 minutes on each side for medium-rare. While the burgers cook, arrange the rolls, cut sides up, on a foil-lined baking sheet and sprinkle with parmesan; broil until toasted.
Transfer roll bottoms to plates. Place burger patties on top, smother with onions, sprinkle with chives and add roll tops.