In a large bowl, cream the butter and sugar together until light and fluffy. Add the eggs to the butter mixture one at a time, fully integrating the first egg before adding the second.
Sift the flour into a large bowl. Add the salt, baking powder, and baking soda and whisk to combine. Add a third of the flour mixture to the egg mixture and beat until fully integrated.
Add half of the buttermilk and mix until fully incorporated. Repeat. Fold in the strawberries. Pour the batter into the prepared loaf pan. Bake for 1 hour.
Arrange the strawberries on a medium-sized baking sheet. Sprinkle the berries with 2 tablespoons of the sugar and place them in the oven alongside the cake. Roast the berries for 45 minutes.
Once the ice cream base is chilled, whisk in the heavy cream. Pour the mixture into an ice cream maker and churn for 30 minutes or until the ice cream resembles soft serve.
Spoon half of the ice cream into a chilled loaf pan. Add 4-5 spoonfuls of the roasted strawberry puree and cut and swirl the puree into the ice cream. Repeat.