In an 8- or 9-inch-wide pan, mix shallots with about 1 teaspoon olive oil. Bake in a 350° oven, stirring occasionally, until shallots are slightly browned and barely tender when pierced, about 40 minutes (20 to 25 minutes in a convection oven). When cool enough to touch, finely chop.
In a wide bowl, mix shallots, 1/2 cup olive oil, vinegar, mustard, and sugar. Peel apples, quarter, core, and cut lengthwise into 1/8-inch slivers. Add to bowl and mix with dressing.
Cut white and red Belgian endive lengthwise into 1/8-inch-wide slices. Add endive and cheese to bowl. Mix gently, and season salad to taste with salt and pepper.