Preheat the oven to 350°F (175°C). Brush a bundt pan thoroughly with melted coconut oil and dust with cocoa powder, then shake out any excess.
In a stand mixer bowl or large mixing bowl, combine the yogurt, 1 1/2 cups sugar, melted oil, eggs, and vanilla. Beat on medium speed until well blended.
In a separate bowl, whisk together the flour, cocoa powder, baking soda, baking powder, and kosher salt until evenly combined.
With the mixer on low, add the dry ingredients to the wet ingredients. While mixing, slowly pour in the hot coffee and mix just until combined; do not overmix.
Pour the batter into the prepared bundt pan and smooth the top. Bake for 50–55 minutes, or until a toothpick inserted into the center comes out with a few moist crumbs.
Allow the cake to cool completely in the pan or on a wire rack before removing and topping with caramel.
Make the caramel: in a medium saucepan over medium heat, cook 1/2 cup sugar, stirring constantly with a heat-resistant spatula or wooden spoon, until it melts into a deep amber liquid. Do not let it burn.
Once the sugar is fully melted, add the 3 tablespoons butter; the mixture will bubble vigorously. Stir until the butter is fully incorporated, about 1–2 minutes.
Slowly drizzle in the 1/4 cup half-and-half (or cream), stirring constantly and carefully as the caramel will bubble up. Continue stirring and cook for about 1 minute more.
Stir in 1/2 teaspoon sea salt, allow the caramel to cool for a couple of minutes, then drizzle evenly over the cooled cake.