I would use a Pivot Table for this.
Say you have this data:
Click inside the table, then click Insert --> PivotTable
It will select the contiguous (non-breaking) range for you, click OK.
It will create a new sheet. Here is where the fun starts.
A new window will appear to the right called "PivotTable Field List." Drag the fields like shown:
Then, click inside the table on any date, then click "Group"
Select "Months" and "Years" then click OK. Now the data is grouped by the year-month like shown:
Go to the PivotTable Field list, click on "Sum of TaT" and instead of SUM, then click "Value Field Settings". Then change the "Summarize Values By" from Sum to Min like so:
Now it shows your min TaT for each month.
Tada! PivotTables makes life EASY :)