1. Rinse foie gras; pat dry. Gently pull apart the lobes. Discard any tough membrane. Cut pieces crosswise into 3/4-inch-thick slices; cover and chill.
2. Divide butter between 2 nonstick frying pans, 12 to 14 inches wide; melt over medium-high heat. Stir 1/4 cup sugar into each. Lay apples equally in pans. Cook until wedges are browned and soft but still hold their shape, 15 to 20 minutes; turn as needed.
3. Set a slice of toast on each dinner plate; arrange apples equally beside toast and place in a 150° oven.
4. Quickly rinse and dry frying pans; place over high heat. When hot, fill with a single layer of foie gras; pieces can touch but shouldn't be crowded. Brown on 1 side, about 30 seconds; turn and brown other side, about 30 seconds more. As slices are browned, quickly put on toast and keep warm. To reduce spattering, pour fat from pan as it accumulates; reserve for other uses.
5. Pour all but 3 tablespoons drippings from 1 pan. Add broth and vinegar to pan and bring to a boil over high heat. Pour hot dressing over greens, mix, and mound beside apples and toast. Serve at once, adding salt to taste.
Handling and cooking foie gras: Keep foie gras chilled until you use it - up to a week sealed in its wrapper. (Foie gras gets soft and difficult to handle when warm.) To sauté foie gras, you need a hot pan and good ventilation, because the rendered fat spatters and smokes. Pour off the fat as it accumulates and use oven mitts to protect your hands.
Nutrition information is not available, but foie gras is very high in fat.