Place a fine-mesh strainer over a large bowl. Whisk the milk, half and half, sugar, arrowroot starch, eggs, salt and vanilla bean in a pot until smooth. Place the half scraped vanilla bean pod i n the pot. Heat over medium-high heat until simmering, whisking constantly. After a simmer is reached, stir and simmer for 1 minute. Then turn off the heat and pour the mixture through the strainer. Discard the vanilla bean pod.
Place in a heat-safe bowl and let sit for 30 minutes. Place a piece of plastic wrap touching the filling and refrigerate until fully chilled. The mixture will thicken and should be the consistency of pudding.
To make the topping
Melt the chocolate with 1/4 cup heavy cream using a double boiler, stirring until smooth. Add more heavy cream to thin out and stir until smooth. Remove from heat. Place in a bowl to le tcool to room temperature. The mixture will thicken as it cools.
To make the doughnuts
Preheat your oven to 350 degrees F and grease your doughnut pan.
Combine the dry doughnut ingredients in a large bowl, mixing well. In another bowl, whisk the eggs together. Then add the milk, yogurt, oil and vanilla. Whisk until well combined.
Pour the wet mixture into the dry ingredients and stir with a large wooden spoon until just combined, being careful not to overmix (stop when you no longer see dry flour).
Spoon the batter into the doughnut molds, filling to just below the top of each mold, 1/8- to 1/4-inch from the top. Bake for 18 to 22 minutes until lightly golden brown around the edges. A toothpick inserted in the center should come out clean. Let cool in the pan for 5 minutes. Slide a knife around the edges to help loosen them out . Then place on a cooling rack and allow to cool fully before topping.
Slice each doughnut in half with a small serrated knife. Spread the filling on the bottom half. Spread a coating of ganache on the top half and sandwich together. Serve immediately.