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For securely connecting on-premises networks to Azure virtual networks with high availability, recommending ExpressRoute is the most appropriate choice. ExpressRoute offers a private connection between on-premises infrastructure and Azure, bypassing the public internet. This approach enhances security by providing greater reliability and performance, which are critical when maintaining high availability.

ExpressRoute connections come with Service Level Agreements (SLAs) that ensure uptime and availability, making it suitable for mission-critical applications. Additionally, it supports larger bandwidth options compared to typical VPN connections, accommodating varying data transfer needs and ensuring that the connection can handle increased demand without degradation of performance.

While options like VPN Gateway do indeed provide secure connections, they rely on the public internet, which can introduce latency and potential security risks. Azure Load Balancer is primarily used for distributing network traffic within Azure and does not facilitate connections from on-premises networks. Virtual network peering is useful for connecting Azure virtual networks but does not extend to on-premises networks, making it unsuitable for the purpose outlined in the question.

Thus, ExpressRoute stands out as the best solution for securely connecting on-premises networks to Azure virtual networks with high availability.