Prepare the smashed feta. In a large bowl, add the block of feta, olive oil, red wine vinegar, lemon juice, and dill. Using a fork, break the feta apart into small pieces, then mash together with the other ingredients. The mixture should be cohesive, with some feta chunks throughout. Set aside.

Prepare the chopped salad. Add the cucumber, tomato, red onion, and dill to a large bowl. Add lemon juice and salt and mix to combine. Set aside.

Grill the flatbread. Brush the flatbread with olive oil lightly on each side. Grill for a few minutes on each side until warm and grill marks appear (alternatively, you can also use a grill pan over the stove following the same instructions).

Assemble the flatbreads. Add smashed feta on each flatbread, spreading it evenly throughout. Using a spoon, spread the chopped salad on top of the smashed feta. Garnish with extra dill.