How to increase pcb trace width after fabrication
With a 20C rise over ambient the vias could still support 0.9A. Which should be plenty for most microcontrollers.

The parasitic resistance of the via might pose a problem at 1.4nH and some ringing in conjunction with the caps.

The problem is if you are going over 1A, the vias might heat up. For traces over 1A the only solution would be to increase the amount of conductor.

The first thing I would do is make sure that it is the vias that are the problem, if you have a thermal camera or a temperature probe then I would take some measurements of the IC's and make sure they are not the cause of the burning. 

Only the traces that carry more than 1A need to be increased. 

If I were doing this with a 2-layer board, I would find a PCB drill (which can be had online for cheap). and drill out the vias that needed the extra current and solder a larger wire from the top to bottom layers.

If this were a 4-layer board I would not attempt to drill because of the potential for shorting on an inner layer. I would find a large gauge wire somewhere from 30 to 20 and solder them from the bottom to top around the outside edge of the board. This is not an elegant solution and could create noise problems