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 (discard shells along with any clinging leeks). 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.
Meanwhile, halve avocado, then peel 1 half (wrap remaining half tightly in plastic wrap and reserve for another use). Cut into 1/2-inch cubes, then toss gently with tomatoes and dill in a bowl.
Divide soup among 6 shallow bowls and spoon some avocado mixture into each. Serve immediately.
It should be noted that the soup, without half and half, can be made 1 day ahead and cooled completely, uncovered, then chilled (with shelled mussels in soup), covered. Remove mussels and reheat soup over moderately low heat before adding half-and-half and then proceeding with recipe.