Preheat the oven to 325°F. Grease an 8-inch springform pan and line the bottom with parchment paper.
Add the butter to a medium saucepan over medium heat. Cook, swirling occasionally, until the butter melts, foams, and develops golden-brown milk solids with a nutty aroma. Remove from the heat.
Immediately add the chopped dark chocolate, olive oil, cocoa powder, and espresso powder to the warm browned butter. Let sit for 1–2 minutes, then whisk until the chocolate is completely melted and the mixture is smooth and glossy.
Add the sugar and flavorings. Whisk in the granulated sugar, maple syrup, vanilla bean paste, and kosher salt until well combined.
In a separate bowl, whisk together the eggs, egg yolk, and Greek yogurt until smooth.
Slowly whisk about ¼ cup of the warm chocolate mixture into the egg mixture to temper, adding it a little at a time while whisking constantly.
Then slowly whisk the tempered egg mixture back into the remaining chocolate mixture. Continue whisking until the batter is completely smooth.
Pour the batter into the prepared springform pan and smooth the top. Bake for 20–24 minutes, or until the edges are set and the center still has a slight jiggle when gently shaken. (Internal temperature should be 160°F; additional baking may be needed.)
Allow the cake to cool completely in the pan before releasing the springform. The center will continue to set as the cake cools.
Whisk together the cocoa powder and powdered sugar. Dust lightly over the cooled cake just before serving. Add a small pinch of flaky sea salt, if desired.
For the richest, fudgiest texture, let the cake come to room temperature before serving. Serve with vanilla ice cream, lightly sweetened whipped cream, or fresh berries if desired.