You will need a coffee grinder to grind the tea leaves here.
In a mini food processor or in a bowl, blend the butter, honey, ground tea and salt. In a medium bowl, toss the apple with the lemon juice. Squeeze out any excess liquid and pulse or mix the apple into the honey butter.
Spread a scant 1/2 tablespoon of the apple butter onto each slice of pumpernickel bread. Top half of the slices with the smoked turkey and close the sandwiches. Using a sharp serrated knife, trim off the bread crusts. Cut the sandwiches into 4 triangles, transfer to a plate and serve.
Make Ahead: The uncut sandwiches can be covered with a lightly moistened paper towel and plastic wrap and refrigerated overnight. Bring to room temperature before serving.