Special devices: @imsodin is right in suggesting a GriGri.
For the method:
The common trick is to have two "belayers." One attaches the GriGri on their harness as usual. The second person stands facing the primary belayer and pulls hand over hand on the rope (essentially pulling away from the primary belayer, through the GriGri). The primary belayer can help by pulling slack down towards the GriGri from above. This should let you belay most climbers quite quickly. But, as always, if at some point there is a lot of slack between the belay device and the climber, the climber must slow down to allow the belayer to catch up. Taking big falls on static rope with slack is dangerous for everyone.
The two-belayer-one-GriGri method was pretty common at local and regional speed climbing competitions in the US until 2018. It was even used at IFSC comps until 2016. In fact, USA Climbing Rule 7.2.1 stated that all routes must be
belayed from below
and this was the accepted practice.
However: Following an incident in 2018 in which a speed climber took a bad fall that was attributed to this belaying style, USA Climbing removed this rule, allowing auto-belays to be used. Similarly, the IFSC and USA Climbing both use special speed auto-belays for speed climbing competition.
I can't definitively speak to which is safer, and I'm not sure anyone can. Hopefully this doesn't need saying, but this should only be done indoors with express permission of the gym staff. Note that some gyms double-wrap their ropes at the top of the climb--don't speed climb with this setup as the friction is both annoying and creates more heat, which is bad for the ropes.