Remove the beef tenderloin filets from the refrigerator 30 minutes before grilling to allow even cooking.
Preheat the grill to medium-high heat, about 350–375°F.
Brush baguette slices lightly with olive oil on both sides.
Grill the baguette slices for 1–2 minutes per side until golden and lightly crisp. Set aside and season lightly with salt.
Season the beef tenderloin generously with kosher salt and fresh ground pepper.
Grill the beef for 4–5 minutes per side, or until desired doneness is reached. Rest for 5 minutes before slicing.
Slice beef thinly against the grain.
Assemble by placing sliced beef on each crostini, spooning béarnaise sauce over the top, and garnishing with fresh chives.
Serve Grilled Beef Tenderloin Crostini with Béarnaise warm or slightly cooled.