Prepare the Lentil Patties : Add the shallot, garlic, parsley, and Fresno chilies to a food processor. Pulse until finely chopped. Add half of the lentils, cumin, 1/2 teaspoon of salt, and a generous pinch of black pepper to the processor. Transfer the paste to a large mixing bowl. Stir in the breadcrumbs, shredded carrot, egg, and the remaining whole lentils. Mix until well combined. Portion the mixture into 6 equal parts and shape them into round patties. Chill the patties in the fridge for 15–20 minutes to firm up.
Make the Goat Cheese Sauce : In a bowl, whisk together the goat cheese, Greek yogurt, apple cider vinegar, honey, and the remaining 1/2 teaspoon of salt. Adjust seasoning to taste. Chill until ready to use.
Cook the Lentil Burgers : Preheat a nonstick pan or griddle on medium heat and lightly coat it with cooking spray or a drizzle of olive oil . Cook the patties for 3–4 minutes on each side, or until golden brown and heated through. Handle gently as lentil patties can be delicate.
Assemble the Burgers : Toast the burger buns or warm the pitas. Spread a generous dollop of goat cheese sauce on each bottom half. Add a lentil patty, a slice of tomato, and a handful of arugula. Top with the remaining half of the bun or fold the pita over.