Stir chia seeds into 2/3 cup of the reserved bean liquid in a small bowl; let sit for a few minutes until chia seeds start to thicken the liquid.
Meanwhile, place beans, breadcrumbs, vinegar, oregano, pepper, garlic, and pepper flakes in a medium bowl. Stir in chia mixture with a fork until mixture holds together. Taste mixture and adjust seasonings, including salt, if necessary. Divide mixture into 6 equal portions and flatten into 4-inch patties. Following Stovetop or Grill Method, cook burgers. While burgers cook, make Tzatziki. Spread buns with Tzatziki, top with burgers, sliced tomatoes, and serve.
Stovetop Method: Heat 2 tablespoons olive or canola oil in a large (12-inch) skillet over medium heat. Add patties and cook, turning only once, until a crisp brown crust forms on both sides, 8 to 10 minutes total.
Grill Method: Heat all burners of a gas grill on high. Lightly brush burger tops with oil. Place burgers, oiled side down on hot rack, cover and grill until spotty brown, about 3 minutes. Lightly brush tops with oil and turn. Cover and continue to grill until burgers are spotty brown on remaining side, about 3 minutes longer