We present the first fully simultaneous fits to the NIR and X-ray spectral slope (and its evolution) during a very bright flare from Sgr A*, the supermassive black hole at the Milky Ways center. Our study arises from ambitious multi-wavelength monitoring campaigns with XMM-Newton, NuSTAR and SINFONI. The average multi-wavelength spectrum is well reproduced by a broken power-law with $Gamma_{NIR}=1.7pm0.1$ and $Gamma_X=2.27pm0.12$. The difference in spectral slopes ($DeltaGamma=0.57pm0.09$) strongly supports synchrotron emission with a cooling break. The flare starts first in the NIR with a flat and bright NIR spectrum, while X-ray radiation is detected only after about $10^3$ s, when a very steep X-ray spectrum ($DeltaGamma=1.8pm0.4$) is observed. These measurements are consistent with synchrotron emission with a cooling break and they suggest that the high energy cut-off in the electron distribution ($gamma_{max}$) induces an initial cut-off in the optical-UV band that evolves slowly into the X-ray band. The temporal and spectral evolution observed in all bright X-ray flares are also in line with a slow evolution of $gamma_{max}$. We also observe hints for a variation of the cooling break that might be induced by an evolution of the magnetic field (from $Bsim30pm8$ G to $Bsim4.8pm1.7$ G at the X-ray peak). Such drop of the magnetic field at the flare peak would be expected if the acceleration mechanism is tapping energy from the magnetic field, such as in magnetic reconnection. We conclude that synchrotron emission with a cooling break is a viable process for Sgr A*s flaring emission.
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