Heat oven to 350ºF (177ºC) with rack set in middle position.
Arrange bread slices in an even layer on a baking sheet. Using a pastry brush, brush both sides of bread slices with 2 tablespoons olive oil; season each side with salt and pepper.
Transfer sheet to oven; bake until bread is just slightly toasted, about 4 minutes. Flip bread slices and continue baking until second side is just slightly toasted, about 4 minutes. Remove from oven and let crostini cool.
In a medium bowl, whisk together remaining tablespoon olive oil, 1 tablespoon vinegar, and 2 teaspoons horseradish; season with a pinch of salt and pepper. Add arugula and toss to coat; set aside.
Heat a cast-iron skillet or braiser over medium-high for 3–5 minutes. Season steak on all sides with salt and pepper. Add 2 teaspoons vegetable oil to skillet and swirl to coat.
Place steak in skillet and cook, without moving, until deeply browned, 2½ minutes. Flip and cook second side, without moving, until deeply browned, 2½ minutes. Reduce heat to medium and, using tongs, sear both edges of steak for 1 minute. Flip back onto first side and continue cooking, flipping every 1½ minutes until steak reaches desired doneness, 3 minutes for medium-rare (125ºF/52ºC), 4–5 minutes for medium (135ºF/57ºC) and 5–6 minutes for medium-well (145ºF/65ºC). Transfer to a plate or cutting board to rest.
Meanwhile, divide Boursin evenly between crostini (scant tablespoon each) and spread into an even layer. Divide arugula salad between crostini.
Thinly slice steak against the grain (the thinner the better) then divide and arrange over crostini.
Spoon about ½ teaspoon horseradish over each crostini and drizzle with honey.