How to increase voltage rise time for micro controller?
You could run the MCU from an external LDO (Low Voltage Dropout) voltage regulator, design the output voltage to be a  bit less than the minimum input voltage. Or, as Tim suggested in the comments, you could use  something like a buck-boost converter to maintain a constant voltage regardless of input voltage. 

If you, for some reason, need the voltage to vary, you could use an LDO with a  "soft start" circuit (see below example) but it will be considerably more compex most likely, unless you can find a part with that functionality built-in. See, for example, this application note.



As yet another alternative, you may be able to disable the internal reset circuit in the MCU configuration and replace it with an external circuit that doesn't have that nasty characteristic. Often the internal ones are not all that good anyway (in particular they may not reset when they should, but this one has the opposite characteristic). Really rapid changes in Vcc can cause the MCU to malfunction so care is called for if you follow this path.