Melt 4 tablespoons of butter in a large stockpot over medium flame. Add the onion and chopped cucumber and cook, stirring frequently, until onion is translucent and softened, about 5 minutes. Be careful not to brown the onion—reduce heat, if needed.

Add broth and bring to a boil. Reduce heat and simmer uncovered for 30 to 40 minutes, stirring occasionally, until vegetables are very tender. Let cool slightly, then blend until smooth in a blender or food processor (you may need to do it in two batches—our blender handled it in one, though).

In a medium bowl or 2-cup measuring cup, lightly beat egg yolks and whisk in cream to just combine. Set aside.

Wipe out the stockpot with a paper towel, then melt the remaining 2 tablespoons of butter in it over medium heat. Add the flour and cook, whisking constantly, until lightly browned into a blond roux, 3 to 5 minutes. Return blended soup to pot and cook, whisking constantly, until soup is slightly thickened, 3 to 5 minutes.

Ladle out 1 cup of soup. Gradually whisk it into the egg yolk/cream mixture, 1 or 2 tablespoons at a time, until combined. Also working slowly, stir this back into the stockpot of soup. The goal here is to not suddenly scramble the egg yolks, but to blend them into the soup. Over low heat, gently cook the soup for another 3 to 5 minutes—do not let it come to a boil. Taste and season with salt and pepper.

Let soup cool, then refrigerate covered until completely chilled, at least 3 hours and up to overnight.

Ladle chilled bisque into small serving bowls and garnish with diced cucumber and chives. Serve.