1. Put steak in a resealable plastic bag with olive oil and shallot. Spread mixture evenly. Seal bag and chill until ready to cook, at least 1 hour and up to 2 days ahead.
2. Heat grill to high (450° to 550°). Let steak sit at room temp 20 minutes, then remove from bag, scraping off shallots, and sprinkle with 1/2 tsp. each salt and pepper. Grill, covered, turning once, until done the way you like, 8 to 10 minutes total for medium-rare. Transfer to a cutting board and let rest 10 minutes.
3. Melt butter in a small frying pan, then add garlic and thyme and cook until simmering and fragrant, 2 minutes. Pour warm butter mixture over steak and slice thinly. Add more salt and pepper to taste.
*Bavette is often categorized as "flap meat" or "butcher steak," along with a few other cuts taken from the area between the sirloin and the flank. Specifically, it is the bottom part of the sirloin. If your butcher isn't familiar with the term "bavette," ask for sirloin flap meat.
Note: Nutritional analysis is per serving.