Heat 2 tablespoons of the oil in a large frying pan over medium-high heat until shimmering. Add the beans, spread into an even layer, and cook undisturbed until the beans are lightly browned on the bottom, 2 to 4 minutes. Add ½ teaspoon of the salt and 1 teaspoon of the za'atar, and stir to combine. Spread the beans out again and cook, stirring as needed, until golden-brown and blistered on all sides, 3 to 5 minutes more.
Add the remaining 1 tablespoon oil to the pan. Add the chard, remaining ½ teaspoon salt, remaining 1 teaspoon za'atar, garlic, and red pepper flakes. Cook, stirring occasionally, until the chard is wilted, 3 to 5 minutes. Remove the pan from the heat, add the lemon juice, and toss to combine.
Divide the beans and greens among 4 bowls, and top each with a poached egg and more red pepper flakes. Serve warm.