Preheat oven to 325 degrees. ( If using canned peaches remove half of the contents of the can drain and chop into small pieces or pure' in a blender ( If you don't want chunks but you just want the flavor) )
In a medium bowl, cream together the butter and brown sugar. Mix in the lavender, and preserve ( or canned peaches). Combine 3 to 3 3/4 of flour to the mix. Mix well.
Sprinkle board with a little of the remaining flour. Knead for 5 minutes, adding enough flour to make a soft dough. Roll to 1/2 inch thickness. Prick with fork and place on ungreased baking sheets.
Bake at 325 degrees F for 20 to 25 minutes. After baking let stand for two minutes ( while still warm not hot but warm),cut into 3x1 inch strips. Let rest until it's completely cool .