Step 1: Set up your grill for direct grilling and preheat to high.

Step 2: Meanwhile, skewer the garlic cloves on toothpicks. Brush the garlic and onion pieces with 1 tablespoon oil and season with salt and pepper. Place the garlic, onion, and peppers on the grate and grill until the garlic and onions are darkly browned, 3 to 4 minutes per side, and the poblanos are charred black on all sides, 4 to 6 minutes per side, turning with tongs to insure even roasting. Transfer the garlic, onions, and poblanos to a plate and let cool.

Step 3: Unskewer the garlic and trim the roots off the onion. Scrape the burnt skin off the peppers with a paring knife. (It’s OK to leave a few burnt spots: they add color and character.) Halve the chiles lengthwise and scrape out the ribs and seeds. Place the garlic, onion, and peppers in a saucepan with the broth, remaining 1 tablespoon olive oil, lime juice, and chopped cilantro and gently simmer the soup over medium heat until the vegetables are soft, 5 to 10 minutes, adding salt and pepper to taste.

Step 4: Puree the soup in a blender. Add the cream or yogurt and puree in short bursts just to mix. (Don’t blend too much, or the soup will curdle.) If serving the soup cold, strain it into a bowl and let cool to room temperature, then refrigerate. If serving the soup hot, strain it back into the saucepan and simmer for 2 minutes. (Straining is optional, but will produce a finer texture.) Correct the seasoning, adding salt or lime juice to taste: the bisque should be highly seasoned. Note: if a touch of sweetness is desired, you could add a teaspoon of sugar or honey.

Step 5: To serve, ladle the bisque into bowls and garnish with a drizzle of cream or a dollop of yogurt. Sprinkle cilantro sprigs over each bowl of soup and serve. Alternatively, ladle the soup into shot glasses or demitasse cups to make shooters.