Preheat oven to 500 degrees. Wash spinach, leaving some water clinging to leaves. Cook in a large saucepan over medium-high heat, covered, stirring occasionally, until wilted, about 3 minutes. Drain in a colander. Let stand until cool enough to handle. Squeeze spinach to release as much liquid as possible. Finely chop, and transfer to a bowl. (You'll have 1/2 cup packed.) Add ricotta and feta. Season with salt and pepper. Stir in 1 egg.
Melt butter in a saucepan over medium heat. Remove from heat; let solids sink to bottom. Skim foam. Carefully pour off clarified butter, discarding solids. Brush baking sheet with clarified butter.
Cover phyllo with plastic wrap and a kitchen towel to keep it from drying out as you work. Remove 3 phyllo sheets, and layer on baking sheet, brushing each with clarified butter before topping with another sheet. Repeat brushing and layering with remaining 3 phyllo sheets to form a second stack on a work surface; cover with plastic wrap and a kitchen towel to keep from drying out.
Place 6 evenly spaced dollops of spinach filling onto first phyllo stack, leaving a 2-inch border around each dollop. Make a well in the center of each filling mound. Crack 1 egg, and separate, reserving white. Carefully tip egg yolk into well of 1 filling mound. Repeat with remaining eggs.
Whisk reserved egg whites and brush onto phyllo between mounds. Top with remaining phyllo stack. Press between filling mounds to seal. Using a pastry wheel, cut around filling mounds to form six 5 1/2-inch squares. Bake until phyllo is golden but yolks are still runny, 6 to 7 minutes. (Centers should still be soft when touched with the tip of your finger.) Serve immediately.