The Brussels sprouts' nuttiness and mild cabbage flavor mingle with tart apples and shallots in this stovetop preparation. Choose firm, crisp apples that will maintain their shape and a bit of crunch when cooked.
Rinse Brussels sprouts well and pull off any loose or yellowing leaves. Trim the stem ends and then quarter each sprout. Set aside.
Heat a large high-sided sauté pan over high heat. Add shallots to the very hot pan and cook, stirring constantly for 2 minutes. Add apples and 1/4 cup water, scraping any brown bits from the bottom as the water sizzles. Cook until the liquid reduces by half, about 2 minutes. Add Brussels sprouts, remaining 1/4 cup water, 2 tablespoons vinegar, salt, and pepper. Reduce heat to medium, cover, and simmer until the sprouts and apples are tender enough to be pierced all the way through with a fork, stirring occasionally, about 15 minutes.
Uncover, stir in remaining 2 tablespoons vinegar and the leaves pulled from sprigs of thyme. Scrape any bits from the bottom of the pan as liquid sizzles and reduces until nearly gone. Transfer to a serving bowl with any of the remaining liquid and serve immediately.