Heat broiler. Place tomatillos, garlic, and serrano chile in a single layer on a rimmed baking sheet, and roast until tomatillos are soft and browned in spots, about 5 minutes. Turn all items; continue cooking until other side is soft and browned, about 5 minutes more. Remove from heat; let cool slightly.
Transfer baking sheet to a wire rack; let cool completely. Peel garlic; place cloves in the bowl of a food processor fitted with the metal blade. Add tomatillos, serrano, and any accumulated juices along with cucumber, onion, cilantro, stock, lime juice, and salt; blend until mixture is smooth. Add yogurt and the water; process until they are just combined.
Transfer to a large bowl or plastic storage container; cover with plastic wrap. Refrigerate at least 2 hours. To serve, ladle into bowls; garnish with avocado.
Since avocado begins to discolor rapidly when exposed to air, it should be cut at the last minute.