Mix together the strawberries, brown sugar, and balsamic vinegar in a bowl, and allow to sit for 1 to 3 hours, stirring occasionally.

Preheat an oven to 400 degrees F (200 degrees C).

Heat olive oil in an oven-safe, heavy steel or cast-iron skillet over high heat. Sprinkle the filets with salt and pepper on both sides, gently place in the hot skillet, and sear each side until well-browned, 1 to 2 minutes per side. Slide the skillet into the preheated oven, and cook until they start to become firm and are reddish-pink and juicy in the center, about 10 minutes. An instant-read thermometer inserted into the center should read 130 degrees F (54 degrees C). Transfer the steaks to a platter, and tent with foil to rest.

Pour steak juices from the skillet into a small saucepan. Strain the strawberries and discard the liquid. Add the strawberries to the steak juices, bring the sauce to a simmer over medium-low heat, and melt the butter into the sauce, tilting the pan several times to gently incorporate the butter into the sauce. Serve each fillet topped with about 1/2 cup of strawberries and a sprinkle of freshly ground black pepper.