Preheat oven to 200°F. Lightly butter baking sheet.
In large skillet, heat 3 to 4 tablespoons oil over medium heat. Add 3 or 4 hash brown patties and fry until golden brown, 7 to 8 minutes per side. Repeat with remaining patties, adding additional oil if needed. Set on baking sheet and place in oven to keep warm.
In medium saucepan, cook beans over low heat, stirring occasionally, until warmed through, about 5 minutes.
Meanwhile, in hash brown skillet, fry eggs until over-easy to over-medium.
For green sauce, mix all ingredients until smooth in food processer or electric mixer. Pour mixture into small saucepan and bring to simmer over medium-low heat, stirring occasionally. Reduce heat to low and cover, stirring occasionally.
To serve, spoon 2 heaping tablespoons beans onto each hash brown patty. Sprinkle each with 2 tablespoons cheese. Top with 1 egg and sprinkle with additional 2 tablespoons cheese. Pour green sauce over and serve warm.