Preheat the oven to 425 F and a place cast iron skillet on the middle rack.
Combine the eggs, flour, milk, salt, and pesto in a blender.
Blend at high speed until very smooth, about 30 seconds. The batter should be frothy.
Carefully remove the hot skillet from the oven, add the butter, and return the pan to the oven to melt the butter. Do not allow the butter to brown or burn.
Once the butter is melted, remove the skillet from the oven and immediately add the batter. Add the mozzarella and tomato to the center of the batter.
Return the skillet to the oven and bake for 20 minutes. Then, turn the oven off and let the Dutch baby sit for 5 minutes in the residual heat.
Remove the skillet from the oven and sprinkle on the basil leaves. Drizzle on the olive oil to serve.