In a small bowl, combine the smoked paprika, cumin, black pepper, coriander, cinnamon, cardamom, and cloves. Set aside 3 tablespoons for this recipe and store the rest in an airtight jar.
Preheat the broiler. Lightly brush the pita breads with olive oil on both sides. Place them on a baking sheet and broil for 5–7 minutes, flipping halfway through, until golden and crisp. Let cool slightly, then break into bite-sized shards.
Heat a generous drizzle of olive oil in a large skillet or Dutch oven over medium heat. Add the bay leaves and garlic, and sauté for 30 seconds, stirring constantly. Add the chickpeas and baharat spice mix. Cook for another 30 seconds, until fragrant.
Pour in the vegetable broth and bring to a boil. Reduce the heat to a simmer and cook uncovered for about 15 minutes, or until the liquid reduces to a thick sauce. Season with salt and pepper. Remove and discard the bay leaves.
In a small bowl, whisk together the tahini and non-dairy yogurt until smooth. If needed, add a splash of water to thin the sauce to a dolloping consistency.
Melt the vegan butter in a small skillet over medium heat. Let it foam and sizzle, swirling the pan occasionally. Once it begins to turn golden brown and smell nutty, remove from heat and stir in the chopped almonds.
To serve, scatter the crispy pita pieces onto a serving platter or into bowls. Spoon over the hot chickpea mixture, dollop with yogurt-tahini sauce, and drizzle with the browned butter and almonds. Sprinkle with smoked paprika. Serve immediately to keep the pita crisp.