is it possible to monitor the intermediate state of a calculation with
Monitor inside the action argument of a Button? The following Monitor
example without a Button works:

Monitor[Do[i, {i, 10^7}],  Refresh[Pause[0.03]; i, UpdateInterval ->
0.01]]

However, when the calculation is started by a Button only the final
result is displayed, the intermediate state of the calculation is not
monitored:

Button["Run",  Monitor[Do[i, {i, 10^7}],   Refresh[Pause[0.03]; i,
UpdateInterval -> 0.01]]]

Button["Run",
 Monitor[Do[i, {i, 10^7}],
  Refresh[Pause[0.03]; i, UpdateInterval -> 0.01]],
 Method -> "Queued"]

hth,