Combine first 4 ingredients in small bowl; mix with fork until peppercorns are slightly crushed. Season with coarse salt and black pepper. Place small sheet of plastic wrap on work surface; place butter mixture atop plastic. Using plastic wrap as aid, form butter mixture into 11/2-inch-diameter cylinder; wrap tightly and chill. DO AHEAD Can be made 3 days ahead. Keep chilled. Bring to room temperature before using.
Rub steak with 1 teaspoon oil; sprinkle both sides with thyme, then coarse salt and 1/4 teaspoon black pepper. Let steak stand at room temperature 30 minutes.
Heat medium nonstick skillet over medium-high heat. Add steak; cook to desired doneness, 3 to 4 minutes per side for medium-rare. Transfer steak to plate; tent with foil. Add 1 teaspoon oil to skillet; add shallot and sauté until slightly softened, scraping up browned bits. Add wine; boil until liquid is reduced by half, stirring often, about 1 minute. Add broth; boil until sauce is thickened, about 2 minutes. Whisk in 3 tablespoons pink-peppercorn butter. Season sauce with coarse salt and pepper.
Cut steak against grain into 1/3-inch-thick slices. Divide between plates. Top with sauce; serve with puree.