To make the tzatziki, grate the cucumber using a box grater. Wring out the liquid with your hands, then place into a kitchen towel or a double layer of paper towels and wring until no more liquid releases (you want it as dry as possible before mixing with yogurt). Add it to a mixing bowl along with yogurt, olive oil, lemon zest/juice, mint, garlic and salt. Mix, taste for seasoning and let it sit for a few minutes to allow the flavors to meld.
Drain and rinse chickpeas then place in a large mixing bowl. Mash them with a fork, a potato masher, the flat side of a meat pounder, a pastry cutter, a sturdy jar, or your hands. Don't over-mash; retain some larger pieces for a better 'chew' experience.
In a non-stick skillet over medium/low heat, add 1 Tb. avocado or grapeseed oil then toss in carrot, chopped leek (or scallions or shallot). When they start to soften, add garlic, a pinch of Kosher salt and a few grinds black pepper. Cook ~2-3 minutes, until soft and slightly browned. If mixture gets too dry, add 1-2 Tb. of water.
Add veggies to chickpeas, along with 1/4 tsp. Kosher salt, 1/8 tsp. freshly ground black pepper, parsley, 1 Tb. olive oil, flour and Feta. Mix to combine. If it seems too dry, add a little bit of water at a time to get to the right consistency (when you can form a tablespoon-sized amount into a small ball that holds together, it's ready). Divide into 6 or 8 portions, depending on how thick you want them, then press into slider-size patties.
In the same non-stick skillet, over medium heat, add 1 Tb. avocado/grapeseed oil. Place patties in the pan, cook until brown (~2-3 minutes) then flip and cook ~2-3 minutes. They may split/break apart when flipping, but don't worry, just shape back together.
While patties cook, toast the buns, cut side down, in a non-stick pan.
Assemble sliders with lettuce, top with tzatziki and serve immediately.