Preheat the oven to 190 degrees C.
Arrange the bread slices on 2 heavy large baking sheets. Drizzle 3 tablespoons oil over the bread slices. Bake until the crostini are pale golden and crisp, about 15 minutes.
Blend the bell peppers and mayonnaise in a food processor until smooth and creamy. Season the bell pepper mayonnaise, to taste, with salt and pepper. Cover and refrigerate.
Tie the beef with kitchen string to help maintain its shape. Season the beef with salt and pepper. Heat the remaining 1 tablespoon oil in a heavy small frying pan over medium heat. Add the beef to the frying pan and cook until brown and cooked to desired doneness, about 8 minutes for rare. Freeze the beef until it is just frozen (this will make it easier to slice), about 3 hours. Using a large sharp carving knife, cut the beef into thin slices.
Spread the bell pepper mayonnaise over the crostini and top with the sliced beef. Arrange the crostini on a platter. Scatter over the capers, sprinkle with the coarse sea salt, and serve.
Do-Ahead Tip: The bell pepper mayonnaise and sliced beef can be made 1 day ahead. Cover separately and refrigerate.
Alternative: If desired, the beef tenderloin can be substituted with 1 pound freshly sliced roast beef. Since the roast beef is already cooked and can be used as is, it makes a great time-saving alternative.