Generously season the tenderloin with salt and pepper.
Mix softened butter, garlic, and rosemary in a small bowl. Rub this mixture evenly over the beef.
Heat a large skillet over medium-high heat and sear the beef on all sides. Place the seared tenderloin in your slow cooker and pour in the beef broth.
Cook on low for 1 1/2-4 hours, checking regularly after 1 1/2 hours, until the internal temperature reads 135°F for medium-rare.
Let the beef rest under foil for 10 minutes before slicing into 1/2-inch thick pieces.
To prepare the sauce, combine white vinegar, shallots, and garlic in a saucepan. Simmer until reduced by half.
Whisk in Dijon mustard and egg yolks, then slowly drizzle in melted butter, whisking constantly. Cook until the sauce thickens slightly.
Remove from heat and stir in tarragon and optional lemon juice. Season to taste.
Serve slices of tenderloin topped with Béarnaise sauce.