The Elliptic Curve Digital Signature Algorithm (ECDSA) is widely used due to its efficiency and security. As an efficient ECDSA implementation, the windowed Non-Adjacent Form (wNAF) is adopted in various applications such as Bitcoin, OpenSSL etc. However, the wNAF implementation of ECDSA is vulnerable to side-channel attacks such as the Flush+Reload cache attack. Previous research demonstrated that lattice-based methods utilizing the Extended Hidden Number Problem (EHNP) can recover the secret key from the double-and-add chain obtained by side-channel attack using only three signatures, extracting an average of 105.8 bits of information per signature for the secp256k1 curve. It is generally believed that key recovery for two signatures is impossible from the information-theoretic perspective. In this paper, we introduce a new perspective to analyze the effects of various lattice construction techniques on the key recovery success rate. We propose an optimal strategy, based on which we achieve ECDSA key recovery for two signatures for the first time and break the information-theoretic limit. More precisely, we first combine the lattice dimension reduction, recentering techniques to construct a more efficient lattice based on the previous work. Then we perform a detailed analysis of the factors affecting the ratio between the target vector length and the Gaussian heuristic, which is critical to the key recovery success rate using the lattice method. Especially, we identify a crosspoint value such that key recovery via the lattice sieving method is feasible if the number of non-zero digits in the leaked ephemeral keys does not exceed this value. Lastly, we introduce an exhaustive search strategy and a method for selecting signatures with fewer non-zero digits to mount an attack. By applying lattice sieving with predicate, our strategy successfully recovers the secret key of the secp256k1 curve using only two signatures. Furthermore, we demonstrate that our approach recovers a 256-bit key with a success probability of 2.858% by extracting an average of 249.7 bits from two signatures, which breaks the information-theoretic limit significantly with a non-negligible success probability for the secp256k1 curve.