Melt butter in a saucepan over medium-high heat. Add onion, tomato paste, and salt to melted butter. Cook, stirring often, until onion softens and tomato paste turns a deep brick red or brown color.

Add garlic, cumin, and cayenne pepper; cook and stir for 2 minutes. Stir in broth and bring to a boil. Stir in lentils, celery, and carrot. Reduce heat to medium-low and simmer, stirring occasionally, until lentils and vegetables are very tender.

Meanwhile, make lemon-mint yogurt: Grind mint and salt in a mortar and pestle to a paste. Add yogurt, lemon juice, and zest; stir together until combined. Refrigerate until ready to serve.