In a large skillet, melt butter over medium heat. Add endive, and cook, turning occasionally, until browned on all sides, about 5 minutes. Pour stock into skillet; bring to a boil, and cover. Reduce heat to medium-low, and simmer, turning endive occasionally, until just tender, about 20 minutes.
Using a slotted spoon, transfer endive to a paper-towel-lined plate; discard stock. Pat endive dry, and set aside.
Make dressing: In a small bowl, whisk together mustard and vinegar. Season with salt and pepper. Whisking constantly, pour in oil in a slow, steady stream; whisk until emulsified. Gently toss each endive in dressing, coating well. Garnish with parsley, and serve at room temperature; set any extra dressing on the side.