In a large skillet over medium-high heat, melt the butter, being careful not to let it burn. Swirl the pan to evenly coat with butter and add the squash, potatoes, and shallots. Toss and stir over medium-high heat until the vegetables have browned around the edges, 8 to 10 minutes.
Add the rosemary and salt. Add pepper generously to taste (about 10 grinds) and stir well. Add the mustard and broth, stirring to mix in the mustard. Bring the broth to a boil, cover the pan, and reduce the heat to medium low.
After the vegetables have been braising for 10 minutes, check the liquid: it should be almost all absorbed, and the vegetables should be completely tender. If the vegetables are tender but still brothy, raise the heat to high and boil the liquid, uncovered, until it’s reduced to a syrupy glaze. Toss to coat the vegetables and transfer to a serving dish.