Integrating explicit Chain-of-Thought (CoT) into end-to-end spoken dialogue models enhances intelligence but incurs prohibitive latency. While the "Thinking-while-Talking" paradigm alleviates this delay, it fundamentally compromises block atomicity, severing the logical connection between interleaved thought and speech. To address this, we present Dual-Reasoner, employing a Streaming Masking Mechanism underpinned by our Dual-Think-30k dataset to guarantee uninterrupted audio streaming. Crucially, to strictly align the fragmented thinking blocks to service speech generation, we introduce the Atomic-Consistency Restoration framework. To secure comprehensive capabilities in high-difficulty reasoning, this mechanism utilizes a quadruple-constraint system to reconstruct logical atomicity, ensuring that "think" chunks act as a rigorous anchor for "talk" outputs. Experimental results demonstrate that Dual-Reasoner achieves comprehensive reasoning enhancements within ultra-low latency constraints: it elevates the VoiceBench score from 67.24 to 73.41 over the baseline, while significantly reducing the Time-to-First-Audio (TTFA) from 20.35s to 3.65s and the Real-Time Factor (RTF) from 7.04 to 1.05.