Toss cucumbers with 1 tsp. salt in a colander over a bowl. Let stand, tossing or stirring a few times, for 30 minutes. Rinse cucumbers with cold water and drain well.
Melt butter in a large saucepan over medium-high heat. Add onion and cook, stirring often, until softened, 3 to 5 minutes. Add flour and cook, stirring constantly, until blended, about 2 minutes. Stir in cucumbers and slowly pour in broth. Bring to a boil over high heat, stirring often and scraping bottom of pan. Reduce heat to medium-low, season with salt and pepper, partially cover and simmer until cucumbers are very soft, 15 to 20 minutes. Remove from heat and let cool slightly.
Working in batches, carefully puree soup in a blender. Cover and refrigerate until well chilled, at least 2 hours. (Can be made up to this point up to 2 days in advance.)
Just before serving, whisk in sour cream until smooth. Stir dill into soup. Season well with salt and pepper. Divide among 8 bowls and garnish with additional dill, if desired.