Hi Daan,
How breakpoints are set, depends on your debug probe technology. For example the SEGGER J-Link is very fast and is able to reprogram the flash page with a software breakpoint. However, this kind of things requires the target to be halted, programming the flash and then continue.
The other way is to use hardware breakpoints: but this still requires the target to be halted afik.
For an explanation how breakpoints work, see https://mcuoneclipse.com/2012/07/29/software-and-hardware-breakpoints/
The only way to get around this is that the debugger is only able to read/write the RAM memory while the target is running, using some bus free cycles. So you would only be able to do this if your code is running in RAM instead of FLASH.
I hope this helps,
Erich