Heat the oil in a heavy 10-inch oven-proof, non-stick skillet over medium-low heat.
Layer the onions, potatoes and sausages, seasoning with salt and pepper as you go.
Cover and cook for 15 minutes.
While that is going on, beat the eggs together with the milk and thyme in a small bowl.
Once the potatoes feel tender when pierced with a knife, it's time to pour the eggs into the skillet and cook until the eggs are set on the bottom (they will still be runny on the top), about 10 minutes.
Bake in the oven until the eggs are set, about 10 minutes.