Take the beef tenderloin out of the refrigerator and allow it to reach room temperature before cooking, roughly 1 to 2 hours.
Combine the softened butter, minced garlic, chopped rosemary, and chopped thyme in a small bowl. Set aside.
Heat the oven to 425°F (220°C). Cut the beef tenderloin in half crosswise, then trim and tie it if the butcher hasn't already done so.
Dry the beef tenderloin with paper towels to remove excess moisture, then sprinkle salt and pepper on both sides.
Warm the avocado oil in a large cast-iron pan over medium-high heat. When the oil begins to shimmer, place the beef tenderloin in the pan and sear it on all sides for a few minutes, about 10 minutes in total, until a golden brown crust forms.
Spread the butter-herb mixture over the top of the beef tenderloin, then put the entire pan into the oven. If using a probe thermometer, insert the probe before putting the tenderloin in the oven. Roast the beef tenderloin for 15 to 20 minutes, depending on your preferred doneness (see temperature tips above).
Take the beef tenderloin out of the oven and move it to a cutting board. Allow it to rest for 10 to 15 minutes so the juices can redistribute. Then remove the twine and cut into 1-inch thick slices. If desired, serve with horseradish sauce.