I would like to be able to track task completion better, and go back to previous months easier. Now the option is to choose ‘within the last… day/month/week’, but this is not helpful if I want to purely look for completed tasks between 1st June and 15th July or other set date.
Welcome to the Asana Community Forum, @Aurora_Virtanen! This is something we are planning to implement soon. I’ll let you know once I have any updates! 
Hi @Aurora_Virtanen, thanks for your patience here!
I’m delighted to confirm that we have just begun rolling out specific date filters in Reporting and Dashboards 
If you don’t have access to this feature yet, you will in the coming days!