(From America's Test Kitchen) Top blade steaks (also called blade or flatiron steaks) are our first choice, but any boneless roast from the chuck will work. If you end up using a chuck roast, look for the chuck eye roast, an especially flavorful cut that can easily be trimmed and cut into 1-inch pieces. Buttered egg noodles or mashed potatoes make excellent accompaniments to carbonnade. The traditional copper-colored Belgian ale works best in this stew. If you can't find one, choose another dark or amber-colored ale of your liking.
Adjust oven rack to lower-middle position; heat oven to 300 degrees. Dry beef thoroughly with paper towels, then season generously with salt and pepper. Heat 2 teaspoons oil in large heavy-bottomed Dutch oven over medium-high heat until beginning to smoke; add about one-third of beef to pot. Cook without moving pieces until well browned, 2 to 3 minutes; using tongs, turn each piece and continue cooking until second side is well browned, about 5 minutes longer. Transfer browned beef to medium bowl. Repeat with additional 2 teaspoons oil and half of remaining beef. (If drippings in bottom of pot are very dark, add about 1/2 cup of above-listed chicken or beef broth and scrape pan bottom with wooden spoon to loosen browned bits; pour liquid into bowl with browned beef, then proceed.) Repeat once more with 2 teaspoons oil and remaining beef.
Add remaining 1 tablespoon oil to now-empty Dutch oven; reduce heat to medium-low. Add onions, 1/2 teaspoon salt, and tomato paste; cook, scraping bottom of pot with wooden spoon to loosen browned bits, until onions have released some moisture, about 5 minutes. Increase heat to medium and continue to cook, stirring occasionally, until onions are lightly browned, 12 to 14 minutes. Stir in garlic and cook until fragrant, about 30 seconds. Add flour and stir until onions are evenly coated and flour is lightly browned, about 2 minutes. Stir in broths, scraping pan bottom to loosen browned bits; stir in beer, thyme, bay, vinegar, browned beef with any accumulated juices, and salt and pepper to taste. Increase heat to medium-high and bring to full simmer, stirring occasionally; cover partially, then place pot in oven. Cook until fork inserted into beef meets little resistance, about 2 to 2 1/2 hours.
Discard thyme and bay. Adjust seasonings with salt and pepper to taste and serve. (Can be cooled and refrigerated in airtight container for up to 4 days; reheat over medium-low heat.)