Boolean symbolic reasoning for gate-level netlists is a critical step in verification, logic and datapath synthesis, and hardware security. Specifically, reasoning datapath and adder tree in bit-blasted Boolean networks is particularly crucial for verification and synthesis, and challenging. Conventional approaches either fail to accurately (exactly) identify the function blocks of the designs in gate-level netlist with structural hashing and symbolic propagation, or their reasoning performance is highly sensitive to structure modifications caused by technology mapping or logic optimization. This paper introduces BoolE, an exact symbolic reasoning framework for Boolean netlists using equality saturation. BoolE optimizes scalability and performance by integrating domain-specific Boolean ruleset for term rewriting. We incorporate a novel extraction algorithm into BoolE to enhance its structural insight and computational efficiency, which adeptly identifies and captures multi-input, multi-output high-level structures (e.g., full adder) in the reconstructed e-graph. Our experiments show that BoolE surpasses state-of-the-art symbolic reasoning baselines, including the conventional functional approach (ABC) and machine learning-based method (Gamora). Specifically, we evaluated its performance on various multiplier architecture with different configurations. Our results show that BoolE identifies $3.53 \times$ and $3.01 \times$ more exact full adders than ABC in carry-save array and Booth-encoded multipliers, respectively. Additionally, we integrated BoolE into multiplier formal verification tasks, where it significantly accelerates the performance of traditional formal verification tools using computer algebra, demonstrated over four orders of magnitude runtime improvements.