Bring a large pot of salted water to a boil.

Heat ⅓ cup olive oil in a large heavy-bottomed pot or skillet over medium heat. Add eggplant and 1 teaspoon salt. Cook, stirring occasionally, until eggplant is softened and golden brown, 8 to 10 minutes. Transfer to a bowl.

Add remaining 2 tablespoons of olive oil to the same pot. Add ⅔ of the finely grated garlic (about 4 cloves’ worth), the cumin, coriander and crushed red pepper. Cook to toast the spices, 15 to 30 seconds. Add tomato paste and cook, stirring often, until darker in color, about 3 minutes.

Add whole peeled tomatoes (including liquid), ½ cup water and 1 teaspoon salt and stir, scraping the bottom of the pot. Using a wooden spoon, crush the tomatoes until the sauce is mostly smooth. Bring to a boil over high heat, then reduce heat and simmer until sauce comes together, about 20 minutes.

In the meantime, make the yogurt sauce by combining the yogurt, finely chopped mint, remaining garlic (about 2 cloves’ worth) and ½ teaspoon salt. Taste and adjust the seasoning, then chill until ready to serve.

Cook the pasta in the salted boiling water until 2 minutes less than al dente. Reserve 1 cup pasta water, then drain pasta.

Add eggplant to the tomato sauce and continue to cook until eggplant is tender and warmed through, another 3 to 5 minutes. Transfer cooked pasta to sauce along with a ladle of reserved pasta water. Stir to coat and continue cooking until pasta is al dente, 2 minutes. Taste for salt and adjust to your preference.

Divide pasta among bowls and top with 1 to 2 tablespoons of the yogurt sauce and some more fresh mint leaves.