Preheat the oven to 350°F (177°C). Line a 9×13-inch metal baking pan with parchment paper.
Beat softened butter and brown sugar until creamy. Mix in cinnamon, salt, baking soda, and flour until crumbly. Reserve 1 cup of the mixture.
Press the remaining crumb mixture evenly into the prepared pan. Bake for 12 minutes.
Slice the apples paper-thin. Melt the butter in a saucepan. Add both sugars and stir until combined. When the mixture is just about to bubble, add the cream or half-and-half. Stir until smooth and thick enough to coat the back of a spoon.
Arrange the apples over the warm crust. Pour the caramel over the apples. Sprinkle with the reserved crumbs and optional turbinado sugar.
Bake for 30–35 minutes, until the topping is golden brown and the apples are tender when pierced.
Cool for 10–15 minutes. Lift the bars using the parchment and cut into 12 large or 16–18 small bars.