Preheat the oven to 375°F. Toss the strawberries with 1/3 cup of the sugar in a baking dish and roast for 25 to 30 minutes, stirring once, until soft and surrounded by thick, syrupy juices, then let cool for 10 minutes.
In a medium saucepan, whisk together the remaining 1/3 cup sugar, cornstarch, and salt, then whisk in the coconut milk, oat milk, and corn syrup. Cook over medium heat, whisking constantly, for 5 to 7 minutes, until the base simmers and thickens slightly.
Pour the base into a blender and add the roasted strawberries with all their juices, the freeze-dried strawberries, lemon juice, and vanilla. Remove the center cap from the lid, cover the opening with a kitchen towel, and blend until smooth.Transfer to a bowl, press plastic wrap onto the surface, and refrigerate for at least 6 hours or overnight.
Melt the chocolate and coconut oil in the microwave in 20-second bursts, stirring between each, until smooth. Spread it very thin over a parchment-lined baking sheet, freeze for 15 minutes until firm, then break into small shards and keep them in the freezer.
Pour the chilled strawberry base into an ice cream maker bowl that has been frozen for at least 24 hours. Churn for 20 to 25 minutes, until thick and creamy.
Working quickly, fold the chocolate shards into the churned ice cream, saving a small handful for the top.
Spread into a 9x5-inch loaf pan, scatter with the reserved shards, cover tightly, and freeze for 4 to 6 hours, until firm. Let sit at room temperature for 10 minutes before scooping.