Preheat the oven to 350°F. Lightly grease a 10- to 12-ounce oven-safe ramekin.
Make the crumb topping. In a small bowl, combine the flour, brown sugar, cinnamon, and salt. Stir in the melted butter until large crumbs form. Set aside.
In a medium bowl, whisk together the flour, brown sugar, granulated sugar, baking powder, salt, and cinnamon.
Add the vegetable oil, Greek yogurt, milk, and vanilla extract. Stir until just combined, being careful not to overmix.
Pour half of the batter into the prepared ramekin. Spoon the salted caramel into the center, then cover with the remaining batter.
Sprinkle the crumb topping evenly over the batter, leaving the crumbs loose for the best texture.
Bake for 20–24 minutes, or until the crumb topping is lightly golden and a toothpick inserted near the edge comes out with a few moist crumbs.
Let the mug cake cool for 10 minutes while you prepare the glaze.
In a small bowl, whisk together the powdered sugar, milk, and vanilla until smooth.
Drizzle the glaze over the warm mug cake. Finish with an extra drizzle of salted caramel and a sprinkle of flaky sea salt before serving.