Heat oil in 12-inch nonstick skillet over medium-high heat until shimmering. Add onions, thyme sprigs, and 1/2 teaspoon salt; cook, stirring frequently, until onions soften and begin to brown, 6 to 7 minutes. Reduce heat to low, stir in sugar, vinegar, and 2 tablespoons water; simmer until liquid has evaporated and onions are glossy, 5 to 7 minutes. Discard thyme springs and season to taste with salt and pepper. Serve over polenta, sprinkling individual portions with cheese and walnuts.

Makes enough to top 4 servings of polenta