We introduce interlaced R2D2 (iR2D2), a DNN series paradigm for scalable image reconstruction from accelerated non-Cartesian k-space acquisitions in MRI with sensitivity map self-calibration. While unrolled DNN architectures provide robust image formation, embedding non-uniform fast Fourier transform operators within the backpropagation graph becomes impractical to train at large scale, e.g., in 2D MRI with a large number of coils, or for higher-dimensional imaging. To address this scalability challenge, we leverage the R2D2 paradigm as a learned version of the Matching Pursuit algorithm that was recently introduced in radio astronomy for fast large-scale Fourier imaging. R2D2's reconstruction is formed as a series of residual images iteratively estimated as outputs of DNN modules taking the previous iteration's data residual as input. Specific to MRI, precomputed sensitivity maps derived from undersampled data can yield an inaccurate measurement operator, which may adversely affect the performance of iterative algorithms such as R2D2. Thus, we extend the R2D2 framework to iR2D2 by introducing a bespoke interlaced architecture that alternates between two R2D2 DNN series to jointly self-calibrate sensitivity maps and form the MR image. We further enhance iR2D2 to operate as an adaptive solver governed by an error-controlled update condition that enforces a sufficient residual energy descent, a dynamic capability fundamentally incompatible with the predefined forward passes of unrolled architectures. Extensive experiments in simulation and on real data, targeting undersampled radial k-space sampling, demonstrate that iR2D2 significantly improves upon R2D2 and outperforms state-of-the-art benchmarks, delivering scalable, high-fidelity imaging with corrected sensitivity profiles.