Preheat your oven to 350°F (175°C). Lightly grease a 9x13-inch baking dish or a 10-inch cast-iron skillet with butter or cooking spray.
In a medium saucepan, combine the sugar and water over medium-high heat. Do not stir once it begins to boil. Gently swirl the pan occasionally until the mixture turns a deep amber color (approximately 8-10 minutes).
Remove from heat immediately and carefully whisk in the heavy cream (it will bubble vigorously). Add butter and miso paste, whisking until completely smooth. Stir in the vanilla extract and salt. Set aside to cool slightly.
Peel, core, and slice the apples into ¼-inch thick pieces. In a large bowl, toss the apple slices with lemon juice. Add flour, cinnamon, nutmeg, and ginger, tossing until evenly coated.
In a medium bowl, combine the oats, flour, brown sugar, chopped nuts, cinnamon, and salt. Add the cold, cubed butter and use your fingers, a pastry cutter, or two forks to work the butter into the dry ingredients until the mixture resembles coarse crumbs with some pea-sized pieces.
Pour about half of the miso caramel into your prepared baking dish. Add the spiced apple mixture, spreading it evenly. Drizzle another quarter of the caramel over the apples, reserving the remaining caramel for serving.
Scatter the crisp topping evenly over the apples, making sure to cover the entire surface.
Bake in the preheated oven for 40-45 minutes, until the topping is golden brown and the filling is bubbling at the edges. If the topping starts to brown too quickly, loosely cover with aluminum foil.
Allow to rest for 15-20 minutes before serving. Serve warm with a drizzle of the reserved miso caramel sauce and a scoop of vanilla ice cream if desired.