Preheat the oven to 375 degrees F.
Slice the strawberries, then put them in a loaf pan and toss with the 6 T. sugar. Bake for 10 minutes. Strawberries should form a syrup. Let cool for at least 20 minutes.
Put the coconut milk in a large pot and stir well. Measure out 1/2 cup and mix with the cornstarch. Stir to dissolve, then add the cornstarch mixture to the pot of coconut milk.
Add the 6 tablespoons sugar and the salt, then tip the loaf pan with the strawberries and pour syrup over the saucepan and let the syrup drain into the pot with the coconut milk (hold back the strawberries). You should get roughly 6 tablespoons of syrup, but it doesn't matter. You can leave some behind with the strawberries, as it will all get pureed and added later.
Turn heat to medium and cook, stirring often until mixture reaches a gentle boil. Stir and let boil for about 3-5 minutes or until it thickens just slightly (more like cream gravy than pudding). Goya brand may take a minute or two longer than brands with guar gum. Remove from heat and transfer to a large bowl. Stir in vanilla extract.
Puree the reserved strawberries and stir strawberry puree into the coconut milk mixture. Cover and chill for at least 3 hours or until very cold. If you want to speed up the chilling, pour the mixture into a large flat dish (like a 9x13 inch dish). The wider surface area will allow air to hit more of the mixture so it will cool faster. Otherwise, just cool it in the bowl.
Process the mixture in your ice cream maker as directed by manufacturer.