For the dough:
Place the butter, powdered sugar and egg yolks in the bowl of a food processor fitted with the blade. Pulse until combined but still speckled with butter. Add the flour and run the machine just until the dough comes together when you pinch it between your fingers (see photo above). Turn dough out onto a large piece of parchment, knead a few times to bring it all together, and pat it into a disc shape. Wrap well in parchment and chill for about half an hour.
Preheat the oven to 350°F. Remove dough from the fridge and let it rest on the counter for 15 minutes. Sprinkle a bit of flour on your work surface, and roll out the dough with a rolling pin to an approximately 12-inch circle. Transfer the dough very carefully to a 10-inch tart pan, pressing on the dough lightly so it sits snugly against the bottom and sides of the pan. Prick the bottom of the shell all over with a fork. Place a piece of parchment paper over the shell, making sure to cover the edges. Spread plenty of dried beans over the parchment, covering the whole bottom of the tart shell. Bake for 15 minutes this way, then remove the parchment and beans and return the shell to the oven for an additional 10 minutes. Remove from the oven and set aside.
For the filling:
Raise the oven temperature to 375F. Place the eggs, yolks and sugar into the bowl of a stand mixer and beat for about five minutes on medium-high speed, until thick and fluffy. Place the butter and chocolate in a microwave-safe bowl and microwave on high for one minute. Stir to combine thoroughly – the residual heat should completely melt both the butter and the chocolate as you stir. (If not, continue to microwave in short bursts until you can stir the mixture to a complete melt.) Pour the chocolate and butter mixture, which should be just warm, into the mixer with the sugar and egg mixture, and mix until well combined.
Pour the filling into the tart shell. Bake for 5 to 10 minutes, until just set. Let tart cool before serving.