Position an oven rack in the upper-middle position and heat the oven to 425°F (220°C). Put the sausages and vegetable oil in a 9-by-13-inch (23-by-33-cm) metal or ceramic baking dish. Turn the sausages to coat them lightly in oil.

Bake until the sausages have begun to brown and the oil is very hot, about 15 minutes. Meanwhile, make the batter.

In a medium bowl, whisk the flour and salt. Make a well in the center, add the eggs and about half of the milk, and whisk until smooth. Gradually whisk in the remaining milk. The batter should be thin, like heavy cream. Let it stand while the sausages start baking.

Carefully remove the hot baking dish from the oven. Quickly pour the batter around and between the sausages; avoid pouring directly over their tops. Return the dish to the oven immediately.

Bake without opening the oven door until the batter is deeply golden, puffed around the sausages, and set in the center, 22 to 28 minutes. Do not worry if it deflates slightly after leaving the oven.

While the toad in the hole bakes, melt the butter in a medium saucepan over medium heat. Add the onion and cook, stirring often, until soft, browned, and sweet, about 15 minutes. If it begins to stick, lower the heat and add a small splash of broth.

Stir the flour into the onions and cook for 1 minute. Gradually pour in the beef broth while whisking. Add the Worcestershire sauce, Dijon mustard, and black pepper. Simmer, stirring frequently, until thickened enough to coat a spoon, about 3 to 5 minutes. Taste and add salt as needed.

Spoon the onion gravy over portions of the baked sausages and Yorkshire pudding. Serve immediately with peas, green beans, or a crisp salad.