In a large bowl, whisk together eggs, flour, milk, water, salt, pepper, and cayenne until completely smooth. Cover and let rest at room temperature for 1 hour to develop maximum fluff.

Set oven to 400°F (200°C). Place a 9×13-inch baking dish or cast-iron skillet with oil in the oven to get smoking hot (about 10 minutes).

Carefully add sausages to the heated pan, turning to coat in oil. Roast for 10 minutes until lightly golden.

Remove pan, swirl the sausages, then pour rested batter evenly around them. Return to oven and bake 25–30 minutes, undisturbed, until puffed and deep golden.

While pudding bakes, melt butter in a saucepan over medium heat. Add onions and cook until soft and caramelized, about 8 minutes. Stir in flour for 1 minute, then whisk in broth and thyme. Simmer until thickened, seasoning to taste.

Slice the puffed pudding, plate with sausages, and drizzle generously with onion gravy.