Position a rack in the center of the oven and heat the oven to 425°F.
Heat 2 Tbs. of the olive oil in a 5-quart Dutch oven or other heavy-duty pot over medium-high heat until shimmering hot. Add the onions and reduce the heat to medium. Cook without stirring until the bottom of the pot begins to brown, 5 to 7 minutes. Sprinkle with 1/4 tsp. salt and stir with a wooden spatula. Continue to cook, stirring and scraping the bottom of the pot frequently and adjusting the heat as necessary, until the onions are well browned, 20 to 30 minutes.
Add the ale and bring to a boil over high heat. Reduce the heat slightly and cook at a vigorous simmer until all but a thin layer of the ale has evaporated, about 2 minutes. Add the chicken and beef broths and bring to a boil. Lower the heat and simmer for 10 minutes to allow the flavors to blend.
Meanwhile, on a rimmed baking sheet, toss the bread cubes with the remaining 2 Tbs. olive oil. Spread the bread in a single layer and bake until golden, 10 to 12 minutes. Remove from the oven, push the bread cubes closely together, and sprinkle with the cheese. Bake until the cheese has melted, about 3 minutes. Let cool slightly.
Season the soup with salt and pepper to taste. Serve with the croutons and a sprinkling of the chives.