Mustard---heat the oil in a small skillet; saute the shallot until soft, about 3 minutes.
Combine the mustard, cranberry sauce, cooked shallots, and vinegar in a food processor; pulse until pureed.
Use immediately or store in the refrigerator for up to 1 week.
To marinate the mushrooms, whisk together the oil, vinegar, marjoram, garlic, and sugar in a bowl; set the clean mushrooms, stem side down, in a glass baking dish; pour the marinade over the mushrooms; cover with plastic wrap and refrigerate for 15 minutes.
Preheat the broiler or prepare a hot fire in a charcoal or gas grill.
Arrange the drained mushrooms on the broiler pan and broil 6 inches from the heat, or place on the grill.
Cook for 3-5 minutes on each side, depending on the size of the mushrooms; set aside to cool slightly.
Lightly toast the cut sides of the split rolls.
Place each on an individual serving plate and spread with cranberry mustard.
Place a grilled mushroom on each roll bottom; sprinkle with salt and a few grinds of black pepper.
Top with the other half of the roll and serve immediately.