To make the Balsamic Teriyaki Sauce, add the butter to a saucepot and set over medium-low heat. When the butter just starts to bubble, add the shallots and the garlic and cook for 3 minutes or until aromatics are soft. Watch to make sure you don't burn the shallots or garlic.
Pour the balsamic vinegar in and bring to a boil. When boiling, immediately turn the heat to medium-low and simmer, uncovered, for 15 minutes or until reduced by half. The balsamic should be think, glossy and coat the back of a spoon. Add the sugar, stock, soy sauce and mirin. Stir well and bring to a boil. When it reaches a boil, immediately turn the heat to low and let simmer for 5 minutes. Let cool and reserve 1/2 cup of the sauce for marinating and the remaining 1/4 cup for serving.
Place the steak and just 1/2 cup of sauce in a large sealable plastic bag. Marinate 2 hours or overnight in the refrigerator. Thirty minutes prior to grilling, remove steak from refrigerator and let it sit at room temperature. Discard the marinade and pat the steak very dry.
Grilling outdoors. If you're grilling outdoors, preheat the grill for direct grilling over high heat. Grilling in pan or stovetop: If you're cooking the steak on the stovetop, set a large frying pan or griddle over high heat. When a bead of water sizzles and evaporates upon contact, add the oil and swirl to coat.
Add the steak to the hot grill or pan in one layer. The pieces should not be touching. Cook for 3 to 5 minutes on each side for medium-rare. Skirt steak varies in thickness, so you might want to add the thicker pieces to the grill or pan first and give them an extra minute head start.
Let the grilled skirt steak rest for 5 minutes, then cut across the grain for ultimate tenderness. Pour the remaining 1/4 cup of the Balsamic Teriyaki Sauce over the meat.