The Question's Actual Proposal Is Basically Impossible
The amount of energy necessary to do this is well defined. You need to apply enough torque to increase angular momentum by a factor of eleven.
This is requires approximately 2.4 * 10^34 joules of energy (there is one factor of eleven that goes into the calculation which isn't present in the linked material).
In round numbers, annihilation of 1 kg of antimatter with 1 kg of matter produces 1.8 * 10^17 joules of energy. So, you would need roughly 10^14 tons of antimatter to accomplish that feet. This is the mass of a typical mountain.
This is unfathomably difficult to do. Even with ultra-extreme technologies that are still scientifically possible, assembling this much energy is for practical engineering purposes impossible, no matter what means you planned on using to apply it.
More Viable Extreme Engineering Alternatives
Build A Death Star-Like Space Station From Scratch
It would be more viable from an engineering perspective to create a huge death star scale man made planet from scratch than the shift the rotation of the planet that much.
Moderate Extreme Temperatures With Heat Exchange Via Wind
If you want to make the planet habitable, you'd be better off for a tiny fraction of the energy budget involved, to set up a megastructure functionally equivalent to giant fans to get an atmosphere to circulate rapidly around the planet carrying heat from the hot side to the cold side to balance it out. Moving a comparatively light atmosphere that was perhaps 60 miles thick and would retain its angular momentum once set spinning except to the extent that friction slowed it, would lead to a much more manageable energy budget to the planet.
Side Note: The Planet Wouldn't Behave The Way You Have Assumed It Would
Indeed, plain old thermodynamics would probably set some of these winds in place even without intervention and would significantly alter your hot desert/ice hell/terminator line conceptualization of the climate zones of the planet with an Earth-like atmosphere that the original question contemplates, so the settler's terraforming would only need to give these natural global prevailing winds a little nudge. You only get that kind of clean three part division on a planet with no atmosphere or a very thin atmosphere. Fluids naturally moderate extremes of temperature.
Mirrors And Energy Generating Filters That Power The Giant Fans
Alternately, or in addition, the settlers could set up giant mirrors to bring some of Proxima's solar radiation to the dark side to heat it up, while erecting some sort of solar filter on the bright side to reduce the heat levels on the hot side while simultaneously generating electricity to power the fans and the community on the surface.
Also, because an atmosphere automatically equalizes temperatures to a significant extent, it isn't terribly important for the mirror part of the terraforming project to be terribly accurately aimed or focused at any particular place.
Your Environmental Impact Statement
Minimization of environmental damage and impact on the life of the
planet as well as its habitability. We don't want to strip the planet
of its atmosphere just to change the day length, do we?
Executive summary: It would be at least as ugly as the scenario in the science fiction movie Avatar, for the local ecology, and probably worse.
The filter and mirror system I suggest would not strip the planet of its atmosphere, but it would utterly screw up any local life adapted to life in the hot and cold areas away from the terminator line, effectively causing almost all of their habitat to cease to exist.
Even recognizing that a planet with that kind of atmosphere wouldn't actually have such an extreme hot/cold/terminator line type climate, the terraforming would still necessarily destroy the habitat of the flora and fauna optimized to spend significant time in the more extreme hot or cold zones of the planet, which the terraforming project would mute somewhat.
