In blender, puree milk, 2 eggs, ⅔ cup water, and ¼ teaspoon salt until smooth. Add both flours and process until smooth, 10 to 15 seconds. Let sit at room temperature 1 hour.
Heat oven to 425°F. On rimmed baking sheet, toss mushrooms with oil and ¼ teaspoon each salt and pepper. Roast, tossing halfway through, until starting to brown, 20 to 22 minutes. Scatter spinach on top and return to oven until starting to wilt, 1 to 2 minutes. Toss spinach with mushrooms and sprinkle with Gruyère. Return to oven until cheese melts, 1 minute.
Meanwhile, heat large nonstick skillet on medium and lightly brush with oil. Stir batter until smooth, then add one-sixth of batter (scant ½ cup) to skillet, swirling to coat. Cook, adjusting heat as necessary, until lightly browned, 2 to 3 minutes. Flip and cook 1 minute more. Transfer to plate. Repeat with remaining batter, brushing with additional oil for each crepe.
Brush oil onto same skillet, add eggs, and cook on medium to desired doneness, 3 to 4 minutes for runny yolks.
Spread 1 tablespoon yogurt on each of 4 crepes (you will have 2 left over), then top with mushroom mixture and fried eggs. Sprinkle with scallions and Parmesan if desired, then fold crepes.
