Slow-Roasted Filet of Beef and Basil Parmesan Mayonnaise

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1 whole filet of beef tenderloin, trimmed and tied (4 1/2 pounds)

3 tablespoons good olive oil

4 teaspoons kosher salt

2 teaspoons coarsely ground black pepper

10 to 15 branches fresh tarragon

Basil Parmesan Mayonnaise, for serving, recipe follows

Basil Parmesan Mayonnaise:

2 extra-large egg yolks, at room temperature

3 tablespoons freshly squeezed lemon juice

1/2 cup freshly grated Parmesan cheese

1 tablespoon Dijon mustard

1/2 cup chopped fresh basil leaves, lightly packed

1/2 teaspoon minced garlic

Kosher salt and freshly ground black pepper

1 cup vegetable oil, at room temperature

1/2 cup good olive oil, at room temperature