Place the 2 tsp. olive oil in a medium-sized pot set over medium heat. Add the onion and garlic and cook until softened, about 3 minutes. Mix in the remaining ingredients, except steaks and 1 Tbsp. olive oil.
Bring sauce to a very gently simmer, partially cover, and cook 15 minutes, stirring occasionally. Cool the sauce to room temperature. Transfer to a tight-sealing jar and refrigerate until needed. (You won’t need all the sauce for the steaks, but the rest will keep for up to 2 weeks and can be used on other items, such as burgers, chicken and pork.)
Preheat your barbecue or indoor grill to medium-high. Brush the steaks with the 1 Tbsp. olive oil; season with salt and pepper. Grill the steaks to desired doneness, allowing 2–3 minutes per side for rare, and 3–4 minutes for medium-rare to medium.
Slather each steak with barbecue sauce, to taste. Let the sauce heat through a minute or so and then serve.