Dual-energy computed tomography (DECT) utilizes separate X-ray energy spectra to improve multi-material decomposition (MMD) for various diagnostic applications. However accurate decomposing more than two types of material remains challenging using conventional methods. Deep learning (DL) methods have shown promise to improve the MMD performance, but typical approaches of conducing DL-MMD in the image domain fail to fully utilize projection information or under iterative setup are computationally inefficient in both training and prediction. In this work, we present a clinical-applicable MMD (>2) framework rFast-MMDNet, operating with raw projection data in non-recursive setup, for breast tissue differentiation. rFast-MMDNet is a two-stage algorithm, including stage-one SinoNet to perform dual energy projection decomposition on tissue sinograms and stage-two FBP-DenoiseNet to perform domain adaptation and image post-processing. rFast-MMDNet was tested on a 2022 DL-Spectral-Challenge breast phantom dataset. The two stages of rFast-MMDNet were evaluated separately and then compared with four noniterative reference methods including a direct inversion method (AA-MMD), an image domain DL method (ID-UNet), AA-MMD/ID-UNet + DenoiseNet and a sinogram domain DL method (Triple-CBCT). Our results show that models trained from information stored in DE transmission domain can yield high-fidelity decomposition of the adipose, calcification, and fibroglandular materials with averaged RMSE, MAE, negative PSNR, and SSIM of 0.004+/-~0, 0.001+/-~0, -45.027+/-~0.542, and 0.002+/-~0 benchmarking to the ground truth, respectively. Training of entire rFast-MMDNet on a 4xRTX A6000 GPU cluster took a day with inference time <1s. All DL methods generally led to more accurate MMD than AA-MMD. rFast-MMDNet outperformed Triple-CBCT, but both are superior to the image-domain based methods.