Make the burgers: Combine 2 cups water, the lentils, rice, 1 teaspoon salt and a few grinds of pepper in a medium saucepan over high heat. Bring to a boil, then cover and reduce the heat to low. Cook until tender, 40 minutes. Remove from the heat and let sit, covered, 10 minutes. If there is water left in the saucepan, cook, uncovered, until evaporated.
Meanwhile, heat the olive oil in a large skillet over medium-high heat. Add the onion and carrot and cook, stirring, until browned, 8 to 10 minutes. Reduce the heat to medium and add the kale, garlic, cumin, curry powder and allspice. Cook, stirring, until softened, about 2 minutes. Add 2 tablespoons water and cook, stirring, until the liquid evaporates. Season with salt and pepper.
Pulse the oats in a food processor until finely ground with some coarse pieces. Add the warm lentil mixture and the vegetable mixture and pulse until combined but still chunky. Season with salt and pepper. Let cool. Form into four 4-inch patties (about 1/2 inch thick). Transfer to a baking sheet and freeze until firm, about 30 minutes.
Meanwhile, make the sauce: Combine the lemon juice and garlic in a small bowl; let sit 5 minutes. Whisk in the tahini, 2 tablespoons water and the hot sauce, thinning with more water, if needed. Season with salt and more hot sauce to taste.
Preheat a grill to medium high. Brush the burgers with olive oil on both sides. Grill until well marked, about 3 minutes per side. Grill the pitas until warmed. Serve the burgers in the pitas with lettuce, tomatoes and pickles and the tahini sauce.