Toast the pistachios in a small baking pan in the oven for 10 minutes or until lightly browned. Remove from the heat, pour the nuts onto a plate to cool them, chop them coarsely, and set aside.

In a small saucepan, mix together the sugar and 1 tablespoon water over medium-high heat. Cook, stirring occasionally, until the sugar caramelizes to a mahogany brown color. Remove from the heat and immediately add the vinegar, standing back to avoid splatters. Whisk briefly, pour into a small heatproof container, and set aside.

Heat a wide, heavy sauté pan over medium-high heat. When the pan is hot, add half of the oil and half of the butter. Before the butter melts completely, add half of the brussels sprouts and season liberally with salt. The pan should be full but should only contain a single layer of sprouts. Sauté for 10 minutes, turning the sprouts so all sides brown evenly. When just cooked through and lightly browned, remove the sprouts from the pan and set them aside.

Add the remaining oil and butter to the pan and sauté the rest of the sprouts. After 9 minutes, add the garlic, parsley, and rosemary and cook until the sprouts are done and the herbs and garlic are fragrant, about 1 minute.

Return the rest of the cooked sprouts to the pan and stir to combine. Warm the sugar-vinegar gastrique in the microwave for 15 seconds to loosen up the sugar. Pour it into pan with the sprouts and add the pistachios and a generous pinch of salt. Toss to coat and serve.