For the cakes: In a large bowl, combine the farro, parsley, arugula, thyme, green onions, garlic, flour, breadcrumbs, Parmesan, eggs, salt and pepper. Mix well.
Heat a cast-iron pan over medium-high heat. Add the butter and olive oil. Form the farro mixture into four cakes by packing it into a 1/4 cup measuring cup. Add the cakes to the hot pan and gently flatten with the back of the measuring cup. Cook until the farro crusts and becomes a dark brown, then flip and cook until well-browned on the reverse side.
For the topping: Heat 2 tablespoons olive oil in a nonstick skillet. Crack in the eggs and cook, sunny-side up, just until the whites are cooked but yolks are still runny.
In a small bowl, toss the arugula with 1 tablespoon olive oil and the lemon juice.
To serve, place a farro cake on plate, cover with a fried egg and top with arugula salad. Grate some parmesan over the arugula and finish with a sprinkle of sea salt.