Coarsely grate 1/2 pound of the potatoes and cut the remaining pound into 1/2-inch cubes. Keep both portions ready to add immediately after the onion cooks.
Melt butter in a 4-quart pot over medium heat. Cook onion and celery for 5 minutes.
Add broth, both potato portions, salt and pepper. Bring to a simmer and cook partly covered for 12 to 15 minutes, stirring from the bottom, until the cubes are tender.
Mash about 1/2 cup of the potato cubes against the side of the pot and stir them back through the broth. Add milk and heat gently for 2 to 3 minutes.
Stir in dill and lemon juice off the heat. Divide among 4 bowls, leaving most potato cubes intact.