To make dough: Dissolve yeast and sugar in warm water and let stand until foamy, about 10 minutes.
Meanwhile, combine flour and 1 teaspon salt in a large bowl. Beat 1 egg in a separate bowl and stir in 2/3 cup milk, 2 tablespoons oil and the yeast mixture. Make a well in the center of the flour and pour in the liquid; mix with a wooden spoon, stirring until mixture comes together and forms a soft dough.
Turn out onto a lightly floured surface and knead until smooth and elastic, about 5 minutes, adding flour as needed to prevent sticking. Shape the dough into a ball, place it in a lightly oiled bowl and turn dough to coat. Cover with a damp cloth and set in a warm place for about 45 minutes, or until doubled in size.
To make filling: Bring a large pot of water to a boil. Add greens, return to a boil and cook for 2 to 3 minutes. Drain, refresh in cold water, squeeze out moisture and chop greens finely. Place them in a large bowl and set aside.
Heat 1 tablespoon oil in a large, nonstick skillet over medium heat and fry scallions, fennel, dill and parsley until softened, 2 to 3 minutes. Stir into the greens, along with feta, 1/3 cup milk, 1/4 teaspoon salt and pepper. Mix well, taste and adjust seasonings. Mix in 2 eggs and rice.
To assemble: Preheat oven to 350 °F. Lightly oil a 9-by-13-inch baking dish.
Punch down dough and divide into 2 balls, one slightly larger than the other. Roll out the larger ball on a lightly floured surface into a 12-by-16-inch rectangle. Place in the prepared dish. Press the dough up the sides and spread the filling evenly throughout.
Roll out the remaining dough, place it over the top of the dish and pinch the edges of the top and bottom crusts together. Bake until crusts are crisp and golden, 45 to 50 minutes. Cool on a rack for 10 minutes before serving.