Melt 1 1/2 tablespoons butter with 1 1/2 tablespoons olive oil in a 12-inch frying pan over medium heat. Add onions and stir in 1/2 teaspoon salt. Cover and cook, stirring occasionally, until onions are limp, about 8 minutes. Uncover and sprinkle with sugar. Increase heat to medium-high and stir often until onions begin to brown, 5 to 7 minutes. Add balsamic vinegar and 1 1/2 teaspoons thyme leaves; stir often until liquid has evaporated, 1 to 2 minutes longer.
Rinse steaks and pat dry. Sprinkle both sides lightly with salt, then coat with pepper. Melt remaining 1/2 tablespoon butter with 1/2 tablespoon olive oil in a 12-inch ovenproof frying pan over medium-high heat (divide among two pans if there's not enough room for steaks in one). Add steaks and cook until well browned on the bottom, 4 to 5 minutes. Turn steaks over and cook until beginning to brown on the other side, about 2 minutes. Transfer pan to a 375° oven and bake until medium-rare (still pink in the center; cut to test), 7 to 8 minutes, or until as done as you like (steaks will continue cooking for a few minutes after you take them out of the oven).
Transfer steaks to warm plates. Spoon onions over the top and sprinkle with remaining 1 1/2 teaspoons thyme leaves.
* Percent Daily Values are based on a 2,000 calorie diet. Your daily value may be higher or lower depending on your calorie needs.