In a large bowl, add the beans, herbs, flour, oregano, mustard, garlic, pepper, salt, and egg. Use your hands to mix everything together, mashing most of the beans to form a pasty mixture (keep some beans whole). Mix in the feta cheese. Form into 6-8 medium sized patties for burgers (or make small patties, the smaller you make them the easier they will be to flip). Make sure the patties are nice and compact.
In a medium bowl with a lid (or a container) add the cucumbers, vinegar, oil, oregano, and salt and mix to combine. Let sit for at least 10 minutes, but overnight would be best.
In a large skillet over medium heat, add 2 tablespoons oil. Once the oil is heated up, carefully place half of your patties into the pan and press down gently with a spatula to flatten the bottoms. Cook for 3-5 minutes or until a deep golden brown. Very carefully flip them over and cook for another 2-3 minutes or until golden. Transfer to a paper towel lined plate. Carefully wipe out the skillet and repeat with remaining 2 tablespoons oil and patties.
Add your veggie burgers to a toasted bun and top with your desired toppings, making sure to put the marinated cucumbers on top somewhere in there. Serve with a simple side salad.