1 medium cucumber (about 8 ounces), cut crosswise into 24 thin slices
1 cup ice cubes
½ (16-ounce) loaf very thinly sliced seedless Jewish rye bread (such a Pepperidge Farms)
1 tablespoon butter, at room temperature
1 (3-ounce) package cream cheese, softened
3 tablespoons mayonnaise
1 small cucumber (about 8 ounces), peeled, finely grated, and drained
1 teaspoon grated onion
1 teaspoon finely chopped fresh dill
Sprigs fresh dill
Combine water, vinegar, and salt in a small bowl, stirring until salt dissolves. Add cucumber slices and ice cubes. Set aside.
Cut each of 8 bread slices into 3 rounds using a 2-inch cookie cutter. Reserve remaining bread for another use. Spread butter evenly over bread rounds; cover with a damp cloth, and set aside.
Remove cucumber slices from water; drain well on paper towels.
Combine cream cheese and mayonnaise in a small bowl, beating with an electric mixer at low speed until smooth and creamy. Add grated cucumber, onion, and chopped fresh dill; mix well. Spread 1 teaspoon cream cheese mixture evenly over buttered side of each bread round. Top each round with slice of reserved cucumber and a sprig of fresh dill.
Cover sandwiches with a damp cloth and refrigerate until ready to serve.