Begin by making the cream cheese spread. Mix together the cream cheese and the mayonnaise with an electric mixer until thoroughly blended.
Add the dill and the chives and stir until completely combined. Cover and set aside.
Cut the cucumber into very thin slices, about 1/4 of an inch thick. Set them aside.
Now assemble the sandwiches. Spread about 1 teaspoon of the cream cheese mixture onto one side of 2 pieces of bread.
Place the cucumber slices on one of the cream cheese-sides of a piece of bread, then place the cream cheese-side of the other piece of bread on top, so that the cucumbers are wedged between the cream cheese sides of the bread slices.
Cut the sandwich into quarters, cutting diagonally across the bread from corner to corner.
Repeat this process with the remaining ingredients. Serve immediately and refrigerate any leftovers.