Wash the cucumber well, peel if needed, and chop it into very small, even pieces. Peel and grate the carrot finely so it blends easily into the mixture. Add both vegetables to a medium mixing bowl.

Add mayonnaise to the bowl along with salt and black pepper. Mix slowly using a spoon until all vegetables are evenly coated. The mixture should be creamy, smooth, and easy to spread. Adjust mayonnaise if needed for consistency.

Place the bread slices on a clean surface. Spread a thin layer of butter on one side of each slice if using. This helps improve taste and prevents sogginess.

Take one slice of bread and add a generous amount of the prepared filling. Spread it evenly using the back of a spoon, making sure all edges are covered.

Place the second slice of bread on top and gently press to secure the filling. Do not press too hard to avoid squeezing the mixture out.

Use a sharp knife to cut the sandwich into halves or triangles. Cut slowly and carefully for neat edges.

Serve immediately for the best texture and flavor.