Heat oil in a medium skillet. Add sliced turnip and sauté until they start to become tender, then add in mushrooms, onion and more oil if needed and continue to sauté until mushrooms are tender and slightly golden brown. Lower heat.
In a medium bowl, whisk together the eggs and milk, if using with a pinch of salt. Add egg mixture to the vegetables, tilting pan slightly to ensure even distribution. Add dollops of Fromage Blanc and cook over low-medium heat until eggs begin to set, around 10 minutes.
Finish cooking under broiler for about 3 minutes.