Halve and core the unpeeled apples, then cut into 1/2-inch dice.
Heat 2 tablespoons of the oil in a medium, non-reactive skillet over medium-high heat. Add onions and sauté, stirring, until translucent and brown around the edges, about 4 minutes. Add garlic and sauté and stir 1 minute more.
Add remaining tablespoon oil to the skillet and diced apples. Sauté, stirring frequently, until apples are translucent and lightly browned, 5 to 6 minutes. Add ginger, coriander, mustard, cinnamon, cayenne and brown sugar, and stir until the sugar starts to melt. Add vinegar and bring mixture to a simmer. Cook, stirring frequently, until mixture is thick and liquids are syrupy, 10 to 12 minutes.
Remove chutney from heat and cool. (Chutney can be prepared 1 day in advance. Cool, cover and refrigerate; bring to room temperature before using.)
For toasts, arrange a rack at center position and preheat oven to 350 degrees. Whisk together olive oil and curry powder, and brush each baguette slice on one side with this mixture. Place toasts, oiled sides up, on a baking sheet and bake until just slightly crisp, about 6 minutes.
Spoon some chutney on top of each toast. Using a vegetable peeler, shave thin slices from the piece of Cheddar and garnish each toast with several thin shavings. (Toasts can be prepared 1 hour ahead. Leave at room temperature, uncovered.) Arrange on a serving platter.