3 leeks (white and pale green parts), quartered lengthwise, then sliced crosswise
2 tablespoons unsalted butter
4 (3-inch) sprigs fresh thyme
1 (12-ounce) bottle lager
1 1/2 cups water
3 pounds mussels, scrubbed well and beards removed if attached
1 cup half-and-half
1/4 teaspoon salt
1/4 teaspoon coarsely ground white pepper
1 (8-ounce) firm-ripe avocado
3 tablespoons coarsely chopped fresh dill
Wash leeks in a bowl of cold water, agitating them, then lift out leeks and pat dry.
Cook leeks in butter in a wide 5-quart heavy pot over moderate heat, stirring occasionally, until softened, 5 to 7 minutes. Add thyme and cook, stirring, 1 minute. Stir in beer and water, then increase heat to moderately high and bring to a boil.
Add mussels and return liquid to a boil, partially covered. Cook, completely covered, stirring occasionally, just until mussels open wide, checking frequently after 4 minutes and transferring to a large bowl. (Discard any mussels that remain unopened after 8 minutes.)
Working over a bowl, remove mussels from shells and put in bowl and discard shells. Pour any cooking liquid accumulated in bowl back into pot. Pour cooking liquid through a sieve lined with a double layer of cheesecloth or dampened paper towels into a 2-quart heavy saucepan. Stir in half-and-half, salt, and white pepper, then heat over moderately low heat until hot (do not let boil). Stir in mussels and heat until just warmed through.
Peel avocado and cut into 1/2-inch cubes, then toss gently dill in a bowl. Divide soup among 6 shallow bowls and spoon some avocado mixture into each. Serve immediately.