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Make the crêpe batter
Combine milk, buckwheat flour, all-purpose flour, 1/2 cup water, eggs, butter, and salt in a blender, and process until smooth, 20 to 30 seconds, stopping to scrape down sides as needed. Cover and chill at least 2 hours or up to 24 hours.
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Make the miso-caramelized onions
Heat butter and oil in a large skillet over medium-high until butter is melted and beginning to foam, 1 to 2 minutes. Add onions, and cook, stirring occasionally, until onions begin to soften, 8 to 10 minutes. Reduce heat to medium, and cook, stirring often, until onions are very soft and caramelized, about 1 hour. Whisk together Shaoxing wine, miso, and gochugaru in a small bowl until miso is dissolved. Pour miso mixture into skillet, and cook, stirring constantly, until wine is fully reduced, 2 to 3 minutes. Remove from heat. Cover and keep warm.
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Cook the crêpes
Heat a 12-inch nonstick skillet over medium-high. Stir crêpe batter. (It should be the consistency of half-and-half. If necessary, stir in additional milk to thin batter.) Lightly grease skillet with butter, and add 1/4 cup crêpe batter. Quickly swirl skillet to evenly coat bottom. Cook until sides of crêpe release from skillet and bottom is lightly golden, about 1 minute and 30 seconds. Flip crêpe, and cook until lightly browned, about 1 minute. Transfer crêpe to a plate. Repeat process with remaining batter to yield 10 crêpes, buttering skillet between batches. Keep crêpes covered with a clean kitchen towel to keep warm. Wipe skillet clean.
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Make the fried eggs
Add 2 tablespoons butter to cleaned skillet, and heat over medium-high until foamy. Crack 4 eggs into skillet, spacing apart evenly. Sprinkle eggs evenly with 1/4 teaspoon salt. Cook until whites are set with golden edges and yolks are runny, 2 to 3 minutes, basting tops of whites occasionally with butter. Transfer to a plate. Repeat process with remaining 2 tablespoons butter, remaining 4 eggs, and remaining 1/4 teaspoon salt.
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Assemble the crêpes
Working with 1 crêpe at a time, spread 2 heaping tablespoons miso-caramelized onions along center of crêpe. Top onions with 1 1/2 tablespoons crème fraîche and 1 egg. Sprinkle with 1/2 teaspoon furikake. Fold right side of crêpe over filling at an angle; fold left side over top, forming a triangle with a point at the bottom and egg peeking through top. Transfer crêpe to a serving plate. Repeat process with remaining crêpes, onions, crème fraîche, eggs, and furikake. Serve immediately.