Melt the butter in a large skillet over medium-high heat. Crack eggs into the melted butter and cook to desired firmness, 5 to 7 minutes.
Heat 1 tablespoon olive oil in a separate skillet over medium heat; cook the garlic and jalapeno pepper in the hot oil until fragrant, about 2 minutes. Stir the white beans, 2 tablespoons olive oil, oregano, and turmeric into the garlic mixture; cook and stir until the olive oil is mostly absorbed, about 5 minutes. Stir the pesto into the mixture and remove the skillet from the heat; season with salt. Top each slice of rye toast with about half of the bean mixture. Squeeze the lemon wedge over the beans. Place one of the fried eggs atop each portion and sprinkle with Parmesan cheese to serve.
