In an electric mixer fitted with a dough hook, combine the flour, milk, salt, and oil. Mix on medium speed for 3 minutes. If needed, add 1 tablespoon warm water to help the dough come together into a solid mass. Continue to knead the dough on medium speed until smooth, about 8 minutes. Wrap in plastic and refrigerate for at least 1 hour, or up to 3 days.
Divide the dough into 6 portions. Using a pasta machine, roll each portion, decreasing the thickness after each pass, into long, very thin 5-inch-wide sheets (setting #1 on most machines). You will need to do this in several batches. Trim the sheets into approximately 10- to 12-inch lengths. Transfer to a tray in between layers of parchment paper and refrigerate (or freeze until needed).
For the Topping
In a medium bowl, whisk to combine the fromage blanc, crème fraîche, flour, oil, egg yolk, and nutmeg until smooth. Season with salt and pepper and keep chilled.
Brown the butter in a large sauté pan over high heat. Add the mushrooms and season with salt and pepper. Sauté until tender, about 5 minutes. Drain onto a paper towel–lined plate.
If you have a pizza stone, place it on the bottom of the oven; otherwise, an upside-down baking sheet can be substituted. Preheat the oven to 500°F.
Place a sheet of dough on a lightly floured pizza peel. Spread the cream topping in a thin layer over the dough, making sure it is evenly distributed. Leave a ¼-inch border around the edge of the dough. Sprinkle one-sixth of the onion, mushrooms, oregano, and chives evenly over the cream topping. Repeat to make all 6 tartes. Slide the flambées, one at a time, onto the pizza stone and bake until crispy on the bottom and lightly browned on top. Slice and serve immediately.