How can I make a 10 minute and 15 minute buzzer timer?
You could use a crystal (lets use 20kHz in this example). Connect a 32bit counter to the crystal and check when the output is equal to 12000000 (=60*10*20000; 10 minutes) or 18000000 (15 minutes).

So for the 10 minute buzzer:
The output of the counter should be 1011 0111 0001 1011 0000 0000
Add logic to the output of the counter to see if the output equals to this binary number. As in: add a inverter on the line where the corresponding bit is 0 and a buffer if the corresponding bit is 1. Add those lines together to an AND-port, if this output is 1, then the 10 minute mark is passed. Be aware that this 1 is only high for 1 period of the crystal. Use it to start some other logic to let the buzzer buzz.

Similar for the 15 minute buzzer.