
Sauté the aromatics. Heat the oil in a large pot over medium heat. Add the onion and carrot, and sauté for 4 to 5 minutes, until softened. Add the diced potato and cook for 2 minutes more.
Add the spices. Add the garlic, tomato paste, paprika, cumin, salt, and pepper, and stir for another 30 seconds.
Simmer the lentils. Add the lentils and broth, bring to a boil, then reduce the heat to low. Cover and simmer for 20 to 25 minutes, until the lentils and potatoes are very soft.
Blend. Use an immersion blender to blend until smooth. (Note: Turkish home cooks usually blend it completely, though you can make it as smooth or textured as you'd like).
Make the pepper butter. In a small pan, melt the butter over low heat. Add the Aleppo pepper and stir for 20 to 30 seconds, until fragrant.
Serve. Ladle the soup into bowls, drizzle with the pepper butter, and serve with lemon wedges on the side.