10.1.1.1 via 192.168.10.1 dev eth0 default 10.1.1.1 dev eth0
You need to revisit your networking knowledge ... specifically what a gateway is...
You need a router (gateway) on 192.168.10.0/24 (presumably 192.168.10.1 in your example). On this you have a default route to the next hop of 10.1.1.1...
In terms of routing systems on 192.168.10.0/24 have no knowledge of 10.1.1.1 and no ability to influence the route other than the next hop on the same local link network as them (192.168.10.1 in this example).