Prep the corn. Slice kernels off the leftover grilled cobs.
If you have time, scrape the cobs with the back of your knife to catch the milky bits—this adds extra corn flavor.
Sauté aromatics. Warm the olive oil in a medium pot over medium heat. Add onion, a pinch of salt, and cook 4–5 minutes until translucent. Stir in garlic and jalapeño, cooking 30–60 seconds until fragrant.
Spice it up. Sprinkle in cumin and smoked paprika.
Stir for 15–20 seconds to bloom the spices without burning.
Add corn and broth. Stir in about two-thirds of the corn kernels and all the diced potato. Pour in the broth and bring to a gentle boil, then reduce to a simmer.
Simmer until tender. Cook 10–12 minutes, or until the potato is soft when pierced. Taste the broth and season with salt and pepper.
Blend for body. Using an immersion blender, blend the soup until about half-smooth, leaving some texture. (Or transfer 2–3 cups to a blender, blend until smooth, and return to the pot.)
Add remaining corn and cream. Stir in the reserved corn kernels and the cream (or coconut milk).
Simmer 2–3 more minutes to warm through without boiling vigorously.
Finish and brighten. Stir in lime juice. Taste and adjust seasoning—add more salt, pepper, or lime as needed. The flavors should feel balanced: sweet, smoky, and fresh.
Serve. Ladle into bowls and top with cilantro or chives, avocado, a spoonful of yogurt or sour cream, a pinch of chili flakes, and crumbled cheese if you like.
Serve with extra lime wedges.