Pulse the onion and carrots in a food processor until chopped into small pieces. Heat 2 tablespoons butter in a large pot over medium-high heat. Add the onion-carrot mixture, 1 teaspoon dried mint and the cumin seeds. Cook, stirring occasionally, until the vegetables start to soften, about 4 minutes.
Meanwhile, chop the tomatoes. Add to the pot and stir until they start to lose their shape, about 2 minutes. Add 7 cups water, the lentils, quinoa, 1 teaspoon salt and a few grinds of pepper. Cover and bring to a boil, then uncover and reduce the heat to a gentle simmer. Cook until the lentils are just tender, about 25 minutes (thin with 1 to 2 tablespoons water, if necessary). Season with salt and pepper.
A few minutes before the soup is done, finely chop the garlic. Heat the remaining 2 tablespoons butter in a small saucepan over medium heat. Add the garlic and cook until it just starts to brown, 1 to 2 minutes. Stir in the red pepper flakes and remaining 1/2 teaspoon dried mint; remove from the heat.
Divide the soup among bowls. Thin the labneh with 1 to 2 tablespoons water and add a dollop to each bowl. Drizzle the garlic butter on top.