Make the dough: Warm the milk in a small saucepan over low heat until 105 degrees F to 115 degrees F. Transfer to a small bowl; stir in the yeast and granulated sugar. Set aside until foamy, 7 to 10 minutes. Meanwhile, whisk the flour and salt in the bowl of a stand mixer.
Add the yeast mixture, eggs and vanilla to the stand mixer. Mix on medium speed with the dough hook until a shaggy ball of dough forms, 1 to 2 minutes. Mix in the butter, 1 piece at a time. Continue to knead the dough with the mixer, scraping the sides and bottom of the bowl occasionally with a rubber spatula, until smooth and elastic, about 6 minutes. Lightly coat a large bowl with cooking spray and transfer the dough to the bowl, turning to coat. Cover tightly with plastic wrap and let rise in a warm place until doubled in size, about 1 hour.
Make the filling: Combine 3/4 cup milk, the granulated sugar, espresso powder and salt in a small saucepan and cook over medium heat, stirring, until the sugar dissolves and the milk is steaming. Whisk the remaining 1/4 cup milk with the egg yolks and cornstarch in a medium bowl. Gradually whisk the warm milk mixture into the egg mixture, then pour the mixture into the saucepan. Bring to a simmer, whisking constantly, until very thick, about 3 minutes. Strain the custard through a fine-mesh sieve into a bowl, pushing it through with a rubber spatula. Stir in the butter and vanilla until smooth. Place a piece of plastic wrap directly on the surface of the custard and refrigerate 30 minutes.
Put the mascarpone in a medium bowl. Whisk in the chilled custard, one large spoonful at a time, until smooth. Transfer to a piping bag fitted with a 1/4-inch round tip. Refrigerate until ready to use.
Lightly flour 2 baking sheets. Turn out the dough onto a lightly floured surface and gently pat until 1/2 inch thick. Cut out about 32 rounds using a 1 1/2-inch cutter; arrange 1 inch apart on the prepared baking sheets and lightly coat the tops with cooking spray. Cover with plastic wrap and let rise until puffy, about 1 hour.
Heat 2 inches of vegetable oil in a large pot over medium heat until a deep-fry thermometer registers 340 degrees F. Meanwhile, make the glaze: Whisk the confectioners' sugar, Marsala and a pinch of salt in a medium bowl, thinning with more Marsala as needed. Cover and set aside.
Working in batches, fry the bomboloni 30 seconds, then flip and continue frying, flipping occasionally, until browned, about 2 more minutes. Remove with a slotted spoon to a paper towel-lined baking sheet to drain. Let cool.
Insert a chopstick or skewer into the side of each bomboloni and wiggle to form a small pocket. Pipe in the filling and dust the tops with cocoa powder. Drizzle with the glaze.