Connection coalescing, enabled by HTTP/2, permits a client to use an existing connection to request additional resources at the connected hostname. The potential for requests to be coalesced is hindered by the practice of domain sharding introduced by HTTP/1.1, because subresources are scattered across subdomains in an effort to improve performance with additional connections. When this happens, HTTP/2 clients invoke additional DNS queries and new connections to retrieve content that is available at the same server. ORIGIN Frame is an HTTP/2 extension that can be used by servers to inform clients about other domains that are reachable on the same connection. Despite being proposed by content delivery network (CDN) operators and standardized by the IETF in 2018, the extension has no known server implementation and is supported by only one browser. In this paper, we collect and characterize a large dataset. We use that dataset to model connection coalescing and identify a least-effort set of certificate changes that maximize opportunities for clients to coalesce. We then implemented and deployed ORIGIN Frame support at a large CDN. To evaluate and validate our modeling at scale, 5000 certificates were reissued. Passive measurements were conducted on production traffic over two weeks, during which we also actively measured on the 5000 domains.