Preheat oven to 50°C (120°F) to keep finished slices warm.
In a bowl, whisk together eggs, milk, salt, and pepper. Stir in most of the grated Parmesan, reserving some for topping. Pour mixture into a shallow dish.
Heat a non-stick pan over moderate-low heat. Add about 2 teaspoons olive oil.
Dip one slice of bread into the egg mixture, allowing it to soak about 15 seconds per side. Let excess drip off.
Place bread into the pan and cook 2–3 minutes per side until dark golden brown. Sprinkle extra Parmesan on top while cooking for a crisp, cheesy crust. Reduce heat if browning too quickly.
Repeat with remaining slices and keep warm in the oven.
In a separate pan, heat 2 teaspoons olive oil. Add sliced mushrooms and cook until softened and lightly browned. Season with salt, pepper, and chopped chives near the end.
Serve the Savoury French Toast with Mushrooms topped with sautéed mushrooms and extra chives. Finish with more Parmesan and a drizzle of olive oil if desired.