In a medium bowl, combine the softened cream cheese, mayonnaise, dill, chives, lemon zest, 1/4 teaspoon kosher salt, and 1/8 teaspoon black pepper. Mix until smooth and evenly combined. For the best flavor, cover and refrigerate overnight.
When ready to assemble the sandwiches, let the cream cheese spread sit at room temperature for a few minutes so it is easier to spread.
Arrange the cucumber slices on a paper towel-lined plate or cutting board. Season lightly with salt and pepper. Let sit while you prepare the bread.
Heat a skillet over medium-low heat. Add a small amount of butter and griddle the bread slices on one side only, just until lightly golden. Work in batches as needed, adding more butter as necessary.
Let the bread cool briefly. While the bread cools, gently blot the cucumber slices with paper towels to remove excess surface moisture.
Spread the cream cheese mixture onto the soft, ungriddled side of each slice of bread.
Layer cucumber slices over 4 slices of bread, then top with the remaining bread slices so the golden sides face out.
Cut into halves, triangles, or fingers. Serve soon after assembling.