Calculating Time in different reference frames

Consider the following problem. A stick passes with speed v though a breadthless person whose(the stick) proper length is L. How does:

• Stick See the time elapsed on my clock?

• I see time elapsed on my clock?

The second part is easy. In my frame ,the stick is length contracted. So the time elapsed is $$T_{me}= L/v\gamma$$

Now the first part. The stick measures the time L/v for the person to pass from one end of stick to another. The stick measures proper time of L/v. Hence he should see the time elapsed on my clock to be $$T_{me}= \gamma T = \gamma L/v$$.

However if I calculate using Lorentz transformation, I get same answer as in a. Where is flaw in my reasoning of 2? Thanks in advance