Is the thin tapered line at the start or end of a plotted line?
If at the beginning it is most likely due to a too small delay after lowering the pen in the timing setup tab. See http://bit.ly/1iFUjAg
(How did you manage to get your posted image to show?)
With a too low delay the egg and/or pen will start to move before the pen touches the egg.
Make sure you “calibrate” the end of servo travel properly before starting the plot (you only have to do this once)
I set the up position to lift the pen tip approximately 5mm off the egg and the down position to a position where the servo arm clears the pen arm by a few mm. I also set the arm and pen height to a position where the top of the pen arm is perpendicular (90°) to the vertical part of the pen arm when the pen tip rests on the egg surface.
Please report back on progress.