Heat the butter and 1 tablespoon of olive oil in a medium skillet over low heat. Add the shallots, season with salt and pepper, and cook for about 30 minutes, until the shallots are tender and caramelized. Transfer the shallots to the bowl of a food processor with the tarragon, cheese, and ⅓ cup of olive oil and blend until smooth. Season with salt and pepper, to taste.
Meanwhile, place a grill pan over medium-high heat. Rub the steak with the remaining tablespoon of olive oil and grill for 6 to 8 minutes per side, until a golden brown crust forms and the meat is cooked through. Transfer the steak to a cutting board and let rest for 10 minutes. Slice the steaks against the grain and transfer to a platter. Spoon the pesto over the top and serve.