Butter and flour one 9-1/2 inch (24cm) round cake pan. Preheat the oven to 375 degrees F (190C; gas 5).
Melt the butter in a small pan over low heat, remove the heat and cool. Melt the chocolate in a small pan over very low heat, checking and stirring it often. Remove from the heat to cool.
Sift together the flour, the baking powder and the salt onto a piece of waxed or parchment paper.
In a large bowl, whisk together the eggs and the sugar until they are light and pale yellow. Sprinkle the dry ingredients over the eggs and sugar, whisking to incorporate them as you do. Fold in the yogurt and vanilla, then the melted butter.
Pour two-thirds of the batter into the prepared cake pan. Fold the melted chocolate into the remaining one-third of the batter until it is thoroughly combined. Pour the chocolate batter on top of the plain batter that is already in the cake pan and run a rubber scraper through the batter several times to swirl the two batters together
Bake the cake in the center of the oven until it is slightly mounded and your finger leaves a very slight impression when you touch the top of it, about 30 minutes.
Remove the cake from the oven and let it cool in the pan for about 15 minutes, then turn it out onto a wire cooling rack. Serve when it is fully cool, or the following day. If you like, you may dust the top with powdered sugar.