In a large mixing bowl or measuring cup, add 4 eggs, ground mustard, salt, pepper, half and half, and parmesan cheese. Whisk vigorously until completely combined.
Arrange 6 slices of sourdough bread in a single layer in a large baking dish. Pour egg mixture over the bread and soak until absorbed, about 5 minutes.
Heat a large frying pan over medium heat. Add 1 tablespoon of oil. Add soaked bread to the hot pan in a single layer, working in batches if needed. Flip when golden brown and fry until the other side is also golden brown. Remove to a plate and cover with foil to keep it warm.
In the same frying pan, add more butter if needed, melt, and once simmering, crack 3 eggs into the pan to fry. Cook over medium heat until the whites are set.
Top 2 slices of French toast with avocado slices, tomato slices, arugula, and a fried egg.