Compared with single-energy computed tomography (CT), dual-energy CT (DECT) can distinguish materials better. However, most DECT reconstruction theories require two full-scan projection datasets of different energies, and this requirement is hard to meet, especially for cases where a physical blockage disables a full circular rotation. Thus, it is critical to relax the requirements of data acquisition to promote the application of DECT. A flexible one half-scan DECT scheme is proposed, which acquires two projection datasets on two-quarter arcs (one for each energy). The limited-angle problem of the one half-scan DECT scheme can be solved by a reconstruction method. Thus, a dual-domain dual-way estimation network called DoDa-Net is proposed by utilizing the ability of deep learning in non-linear mapping. Specifically, the dual-way mapping Generative Adversarial Network (DM-GAN) was first designed to mine the relationship between two different energy projection data. Two half-scan projection datasets were obtained, the data of which was twice that of the original projection dataset. Furthermore, the data transformation from the projection domain to the image domain was realized by the total variation (TV)-based method. In addition, the image processing network (Im-Net) was employed to optimize the image domain data. The proposed method was applied to a digital phantom and real anthropomorphic head phantom data to verify its effectiveness. The reconstruction results of the real data are encouraging and prove the proposed method's ability to suppress noise while preserving image details. Also, the experiments conducted on simulated data show that the proposed method obtains the closest results to the ground truth among the comparison methods. For low- and high-energy reconstruction, the peak signal-to-noise ratio (PSNR) of the proposed method is as high as 40.3899 and 40.5573 dB, while the PSNR of other methods is lower than 36.5200 dB. Compared with FBP, TV, and other GAN-based methods, the proposed method reduces root mean square error (RMSE) by, respectively, 0.0124, 0.0037, and 0.0016 for low-energy reconstruction, and 0.0102, 0.0028, and 0.0015 for high-energy reconstruction. The developed DoDa-Net model for the proposed one half-scan DECT scheme consists of two stages. In stage one, DM-GAN is used to realize the dual map of projection data. In stage two, the TV-based method is employed to transform the data from the projection domain to the image domain. Furthermore, the reconstructed image is processed by the Im-Net. According to the experimental results of qualitative and quantitative evaluation, the proposed method has advantages in detail preservation, indicating the potential of the proposed method in one half-scan DECT reconstruction.