Preheat the oven to 350°F and grease a 9-inch bundt cake pan. For the chocolate cake, beat together the butter and sugar with a hand mixer until the mixture becomes creamy and turns a light color. Add in the egg and continue to beat the mixture on the highest setting. Add in the flour, baking powder, cocoa powder, baking soda, plain yogurt, and milk and continue beating until the batter becomes creamy and smooth.
Pour the chopped pistachios for the topping into the bundt cake pan and spoon the chocolate cake batter on top. Tap the bundt cake pan on the counter to spread the batter around evenly.
For the flan, beat the eggs, milk, sweetened condensed milk, cream cheese, and vanilla extract on the highest setting until combined. Position a spoon over the bundt cake pan and pour the flan mixture directly onto the spoon so that it gently covers the chocolate cake batter.
Place the bundt cake pan in a large casserole dish and fill the casserole dish halfway with hot water. Place a lid on the bundt cake pan, transfer the casserole dish to the oven, and bake the impossible cake at 350°F for 90 minutes. Once the impossible cake has baked, remove it from the oven and let it cool for at least 2 hours.
For the caramel nest, heat the sugar and water in a saucepan. Slowly caramelize the sugar until it turns a light golden color, then remove it from the heat. Dip a fork into the caramel and drip the liquid caramel in thin lines over a metal or stone surface. Once the caramel has solidified, use your hands to gently mold the caramel into a ring. For the caramel sauce, add the heavy cream to the leftover caramel in the saucepan and bring it to a light boil. Once the caramel sauce has cooked briefly, remove it from the heat and let it cool down slightly.
Flip the impossible cake over onto a serving dish or platter, place the caramel nest around it, and top it off with caramel sauce and some additional chopped pistachios.