How to Make It
Fill a steamer pot with 1 inch of water; add steamer insert (water should not touch bottom of basket). Bring water to a boil over medium-high; reduce heat to medium to maintain a simmer.
Meanwhile, peel 1 3/4 pounds zucchini, and cut into 2-inch-thick pieces. Lay zucchini pieces flat in steamer basket. Cover and cook until zucchini is tender when pierced with a paring knife, about 15 minutes. Transfer zucchini to a medium bowl; let cool 10 minutes.
Transfer zucchini (including any liquid from zucchini accumulated in bowl) to a blender; add buttermilk. Process until smooth, about 25 seconds. Using the bottom of a ladle, push squash mixture through a fine wire-mesh strainer into a large bowl; discard any solids. Repeat straining procedure a second time until soup is very smooth. Stir in 13/4 teaspoons salt. Refrigerate, uncovered, until chilled, about 2 hours.
Cut 5 eggs into 1/4-inch pieces; transfer to a small bowl, and gently stir in 1/4 teaspoon salt. Slice remaining egg lengthwise into 6 slices. Set egg slices aside.
Cut remaining small zucchini into 1/4-inch cubes to equal 1/2 cup. (Reserve any remaining zucchini for another use.) Heat a large cast-iron skillet over high until very hot, about 4 minutes. Add 1 tablespoon oil; swirl to coat. Add zucchini cubes; cook, stirring often, until crisp-tender, about 2 minutes. Transfer to a medium bowl. Add 3 tablespoons oil and potatoes to skillet. Cook, stirring occasionally, until potatoes are tender and browned, 12 to 14 minutes. Let cool 15 minutes. Transfer potatoes to bowl with zucchini, and stir in pickled pattypan squash cubes, chives, dill, remaining 1 tablespoon oil, and remaining 1 teaspoon salt.
Ladle about 3/4 cup chilled zucchini soup into each of 6 shallow bowls. Top servings evenly with potato mixture, chopped eggs, egg slices, and roe. Garnish with dill leaves.