Whisk the chilled non-dairy milk and the cornstarch together in a small bowl or container. Set it aside.
Melt one tablespoon of the vegan butter in a medium pot over medium heat, then add the pecans. Toast the pecans in the butter for 3 to 4 minutes, stirring frequently to avoid burning, until they've darkened slightly and smell nutty. Remove the pecans from the pot and transfer them to a plate.
Add the remaining 4 tablespoons of vegan butter to the pot, along with the brown sugar and salt. Stir and heat the butter and brown sugar until they melt, then come to a low simmer. Reduce the heat to low and let the mixture simmer for 3 minutes, stirring constantly.
Whisk the coconut cream, sugar, and corn syrup into the pot with the brown sugar mixture. Give the cornstarch mixture a quick stir to distribute any cornstarch that has settled, then add that to the pot as well. Raise the heat and bring the mixture to a simmer while whisking frequently. You may see some bits of caramelized brown sugar in the mix — they will dissolve eventually.
Once the mixture comes to a simmer, let it cook for about a minute longer, just until thickened slightly. Remove the pot from heat and whisk in the vanilla.
Let the mixture cool completely, giving it an occasional whisk to prevent a skin from forming (Note 1).
Whisk in the rum, then transfer the mixture to your ice cream maker and churn according to the manufacturer's directions until creamy and partially frozen, but not yet solid, about 25 minutes. You want the consistency to be similar to soft serve.
Add the toasted pecans and churn about 3 minutes longer, until evenly distributed.
Transfer the ice cream to containers and freeze until fully set.
Let the ice cream sit out for 10 to 15 minutes before scooping, or warm it up for a few seconds in the microwave (Note 2).