Actually the second round of FFT is exactly doing that, as by definition differentiation of phase is frequency. In realistic case if target is moving, the target position will not change but its phase will change along the column (also called slow-time, where the row in your case is called fast-time), the second FFT shows this changing phase as Doppler frequency. Furthermore, what second FFT also does it something similar to pulse integration. More the number of rows larger is your integration gain.
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