Step 1

Purée garlic, ginger, chile, chili paste, fish sauce, lime juice, gochugaru, and vinegar in a blender until smooth.

Step 2

Do ahead: Sauce can be made 1 week ahead. Cover and chill.

Step 3

Heat 4 tablespoons oil in a large pot over medium-high heat. Cook onion, carrot, and celery, stirring often, until softened, 8–10 minutes. Add coriander seeds and paprika and cook, stirring, until fragrant, 2 minutes. Add wine, vinegar, and 6 cups water; season with salt and pepper and bring to a boil.

Step 4

Add octopus to liquid, reduce heat, and partially cover pot. Simmer gently, turning octopus occasionally, until flesh is tender enough to cut with a spoon, 60–75 minutes. Transfer octopus to a platter with a slotted spoon; let cool. Rub off skin with paper towels. Separate tentacles and cut into equal lengths.

Step 5

Heat a large skillet, preferably cast iron, over medium-high heat. Toss octopus with 1 tablespoon oil and season with salt and pepper. Cook, turning occasionally, until charred, 8–10 minutes. Brush with chili sauce (you may not use all of it) and cook, turning occasionally, until sauce is deeply caramelized, about 5 minutes. Remove from heat and toss with butter to coat.

Step 6

Toss herbs, lime juice, and 1 tablespoon oil in a small bowl; season with salt and pepper. Swipe yogurt across plates and top with octopus, then herb salad; drizzle with more oil.

Step 7

Do ahead: Octopus can be braised 2 days ahead. Cover and chill.