Preheat the oven to 275 degrees. Use an oven thermometer to be sure your oven temperature is accurate.
Place the filet on a sheet pan and pat it dry (all over) with paper towels. Brush the filet all over with the oil, reserving about half a tablespoon. Sprinkle it all over with the salt and pepper (it will seem like a lot but believe me, it makes a difference). Place the tarragon branches around the beef, tying them in 4 or 5 places with kitchen twine to keep them in place, and then brush the tarragon with the reserved oil.
Roast the filet of beef for 1 1/4 to 1 1/2 hours, until the temperature registers 125 degrees in the center for rare and 135 degrees for medium-rare. Cover the filet with aluminum foil and allow to rest for 20 minutes. Slice thickly and serve warm or at room temperature with Basil Parmesan Mayonnaise.
Basil Parmesan Mayonnaise:
Yield:about 2 cups
Place the egg yolks, lemon juice, Parmesan, mustard, basil, garlic, 1 tablespoon salt, and 1 teaspoon pepper in a food processor fitted with the steel blade. Process for 20 seconds, until smooth. Combine the vegetable oil and olive oil in a 2-cup liquid measuring cup. With the processor running, slowly pour the oil mixture through the feed tube to make a thick emulsion. Taste for seasonings -- the mayonnaise is a sauce so it should be highly seasoned. Store in the refrigerator until ready to use; it will keep for up to a week.