With a backer.Forstener bit
never thought about a router. InterestingA router will do the best job.
With a drill, the best results would be to tape the wood, use a sharp brad-point bit in a press, spin at relatively high speed, move slowly, duct collect/vacuum as you drill, and back up the first sign of chips or clogging. I usually shortcut most of these hoping for the best, but if I really need it clean, I do all six.
A bladed, not toothed forstner bit and clamped backer board is a safe bet, but some of us would use a spiral downcut router bit into a clamped backer board.