Heat 1/4 cup olive oil in a large, heavy saute pan over medium-high heat. Add the squash. Season with some salt and pepper. Saute until slightly browned and softened, 8 to 10 minutes. Add the vegetable broth. Reduce the heat to medium-low and simmer, covered, until cooked through, 15 to 20 minutes. Season with more salt and pepper, and set the squash aside to cool slightly.
Meanwhile, prepare the tangerines and vinaigrette. Peel 3 of the tangerines and slice them into 1-inch rounds, removing any seeds. Place the sliced tangerines in a bowl. To make the vinaigrette, juice the remaining tangerine into a small mixing bowl. Add the remaining 2 tablespoons olive oil and the rosemary. Whisk until combined. Season with salt and pepper.
In a large serving bowl, add the mixed salad greens. Toss the salad with the vinaigrette until evenly coated. Top with the warm butternut squash. Arrange the sliced tangerines on top and sprinkle with the dried cranberries. Serve warm.