Rinse red lentils in a fine mesh strainer under cold water until water runs nearly clear. Set aside to drain.
Peel and dice the onion, carrots, and potato into ½-inch pieces. Mince the garlic cloves.
Heat 2 tablespoons olive oil in a large pot over medium heat. Sauté onion for 5-6 minutes until translucent and edges begin to turn golden.
Add carrots and cook for 3-4 minutes, then add potato and cook for 2 minutes. Add minced garlic and stir for 30 seconds until fragrant.
Add tomato paste to the hot pan and cook for 1-2 minutes until caramelized. Stir in cumin, paprika, dried mint, black pepper, and cayenne (if using) for 30 seconds until fragrant.
Add rinsed lentils and 6 cups of broth. Add 1 teaspoon salt, increase heat to medium-high, and bring to a boil. Reduce heat to medium-low and simmer covered for 20-25 minutes until lentils and vegetables are completely tender.
Remove pot from heat and cool for 3-5 minutes. Blend with an immersion blender until completely smooth and velvety, about 60-90 seconds.
Return soup to medium-low heat. Taste and add remaining ½ teaspoon salt or more as needed. Stir in 2 tablespoons fresh lemon juice. Add warm broth or water if a thinner consistency is desired.
In a small saucepan over low heat, warm 3 tablespoons olive oil for 1 minute. Add Aleppo pepper flakes and stir for 30-60 seconds until oil turns deep red. Remove from heat immediately.
Ladle soup into bowls and drizzle with Aleppo pepper oil. Garnish with fresh parsley and serve with lemon wedges and warm crusty bread.