Combine 2 tablespoons olive oil, the garlic, red wine vinegar, and 1 teaspoon thyme in a zip-top plastic bag. Add steaks, turning to coat, and seal bag. Refrigerate 1 to 4 hours, turning occasionally.

Meanwhile, slice onions ¼ inch thick and separate into rings. Halve tomatoes lengthwise.

Preheat a grill for high heat (450° to 550°). Set steaks on a plate and let stand at room temperature 20 minutes. Season with 3/4 teaspoon salt and ¼ teaspoon pepper.

Place a 12-inch cast-iron skillet on the grill’s cooking grate and preheat with the lid closed for 5 minutes. Add the remaining 2 tablespoons olive oil to the skillet and stir in onions and sugar. Cook, stirring occasionally and closing the grill in between, until onions are softened and starting to brown, 7 to 9 minutes. Stir in tomatoes and cook until lightly wilted, 2 to 3 minutes. Add balsamic vinegar and 1/4 teaspoon salt and cook 1 minute. Remove from heat and stir in 1 tsp. thyme and the butter until it melts. Set pan aside and keep warm.

Following the insertion guide on the Yummly Thermometer app, insert the thermometer horizontally into a steak until the metal is completely covered.

Grill steaks 2 minutes with the lid closed, then rotate 1/4 turn. Grill 2 minutes longer. Turn steaks over (or flip when directed to by the thermometer app) and cook 2 minutes; rotate the steaks 1/4 turn. Grill until steaks are as done as you like, 1 to 2 minutes longer for medium-rare (130°). Transfer steaks to a cutting board and let rest 5 minutes.

Taste onion mixture and add more thyme, salt, pepper, and balsamic vinegar if you like. Slice steaks and serve with the onion mixture.