Heat oil in a Dutch oven on medium. Add celery, leeks, onion, and 1/2 teaspoon salt and cook, covered, stirring occasionally, until tender, 5 to 7 minutes.
Add potatoes, thyme, 1/4 teaspoon each salt and pepper, and 6 cups water and bring to a boil, then simmer 8 minutes. Add asparagus and simmer 2 minutes.
Add sugar snap peas and beans and simmer until vegetables are just tender, 3 to 4 minutes more. Discard thyme sprigs. Sprinkle soup with dill and serve with bread, if desired.