In a large Dutch oven or heavy pot with a tight-fitting lid, melt 1 tablespoon butter over medium-high. Season chicken with salt and pepper, add to pot, and cook, stirring occasionally, until browned on all sides, about 5 minutes. With a slotted spoon, transfer chicken to a medium bowl. Add 2 tablespoons butter, scallions, bell pepper, celery, and carrots to pot. Cook, stirring occasionally, until scallions and celery are soft, about 4 minutes. Stir in thyme and flour and season with salt and pepper; cook 1 minute. Return chicken to pot and whisk in beer. With your hands, roughly tear tomatoes and add to pot along with juices. Bring to a rapid simmer and cook, uncovered, 30 minutes. Season to taste with vinegar.
Reduce heat to a medium simmer and drop dough by rounded tablespoons on top of stew. Cover and simmer until dumplings are cooked through, 7 to 10 minutes.
Use one spoon to scoop the dumpling dough and a second to easily scrape it from the first spoon into the pot.