How do Cosmic Microwave Background missions/telescopes measure CMB temperatures? I understand that CMB has Planck's spectrum. So I was thinking one strategy could be to measure the spectral radiance corresponding to different frequencies coming from a certain direction and then compared these data points with theoretical Planck's law. Is this technique practically feasible?
That is what is done. This is shown in an old xkcd comic https://xkcd.com/54/
The curve shows the distribution of frequencies in the CMB, and by using the marked value of the maximum you can determine the value of T, the apparent (red-shifted) temperature of the CMB