Bring a large pot of salted water to a boil. Melt 2 tablespoons butter with the olive oil in another large pot or Dutch oven over medium-high heat. Add the carrots, onion, thyme and a big pinch of salt to the hot oil. Cook, stirring occasionally, until the vegetables start softening, about 4 minutes. Add the cabbage, 1/2 teaspoon salt and a few grinds of pepper. Cook, stirring occasionally, until the cabbage is tender, about 5 minutes.
Add the gnocchi to the boiling water and cook as the label directs. Reserve 1 cup cooking water, then drain the gnocchi and add to the cabbage mixture. Add 3/4 cup of the reserved cooking water, the corned beef, mustard and remaining 2 tablespoons butter. Reduce the heat to medium and cook, stirring, until everything is coated, 1 to 2 minutes, adding the remaining 1/4 cup cooking water as needed; season with salt and pepper.
Divide the gnocchi among shallow bowls. Top with the chives.