Heat the extra-virgin olive oil in a large pot over medium heat. Add the diced onion and carrots, cooking for 4 to 5 minutes until softened. Stir in the diced potato and cook for an additional 2 minutes.
Add the minced garlic, tomato paste, paprika, ground cumin, kosher salt, and black pepper. Stir continuously for 30 seconds until fragrant.
Pour in the rinsed lentils and vegetable broth. Bring the mixture to a boil, then reduce the heat to low, cover the pot, and simmer for 20 to 25 minutes until the lentils and potatoes are completely soft.
Use an immersion blender to blend the soup until smooth, adjusting texture to preference.
In a small pan over low heat, melt the butter (or warm the olive oil). Stir in the Aleppo pepper for 20 to 30 seconds until fragrant, then remove from heat.
Ladle the soup into serving bowls, drizzle the warm Aleppo pepper butter over each portion, and serve with lemon wedges.