Preheat the grill to medium-high heat (approximately 400°F).
Brush baguette slices with olive oil and grill for 1 minute per side until toasted and marked; set aside.
Season the beef tenderloin generously with salt and pepper on all sides.
Grill the beef tenderloin for 4-5 minutes per side or until the internal temperature reaches 130°F for medium-rare.
Remove beef from the grill and allow it to rest for 10 minutes before slicing into 24 thin medallions.
To prepare the Béarnaise, whisk egg yolks, tarragon vinegar, and minced shallots in a heatproof bowl over a pot of simmering water.
Slowly drizzle the melted butter into the egg mixture while whisking vigorously until the sauce is thickened and emulsified.
Remove from heat and stir in the fresh tarragon and lemon juice.
Assemble each crostini by placing one beef medallion on a toasted baguette slice and topping with a dollop of warm Béarnaise sauce.