For the beignets: Whisk together the flour, sugar, salt and yeast in a large bowl until completely combined.
Whisk together the milk, water and egg in the bowl of a stand mixer until well combined. Add the dry ingredients. Fit the mixer with a dough hook and mix on low speed until a soft and sticky dough has formed, about 2 minutes. Add the shortening and mix on medium speed until the dough is smooth and starts to climb up the hook, about 8 minutes.
Lightly oil a large bowl. Remove the dough from the mixing bowl, gently shape into a ball and place seam-side down in the bowl. Cover the bowl with plastic wrap and let rise in a warm place until doubled in size, 1 to 2 hours, depending on the temperature of the room.
For the raspberry syrup: Meanwhile, combine the raspberries, sugar, lemon juice and 3/4 cup water in a medium saucepan. Bring to a boil over medium-high heat, then reduce the heat and simmer, stirring occasionally, for 5 minutes. Strain the mixture through a mesh strainer, pressing down on the berries with a rubber spatula to extract all the juice. Pour the mixture back into the saucepan and simmer over medium heat until slightly thickened, about 10 minutes. Remove from the heat and let cool to room temperature before using.
When the dough is ready, turn it out onto a floured work surface. Flour your hands and gently press the dough into a rectangle. Lightly flour the surface of the dough and a rolling pin, then roll the dough into an 8-by-10-inch rectangle. Using a floured bench scraper or chef’s knife, cut the dough into twenty 2-inch squares. Let the squares rest for 10 minutes.
Meanwhile, pour enough oil into a large Dutch oven to come 2 inches up the sides. Clip a deep-fry thermometer to the pot and heat over medium-high heat until the thermometer reaches 360 degrees F.
Working in batches, fry the dough squares, turning once and adjusting the heat to maintain temperature, until golden brown, about 2 minutes. Transfer to a clean wire rack set over a rimmed baking sheet to drain.
Pile the beignets on a serving platter, dust with the confectioners’ sugar, and drizzle with some raspberry syrup. Serve warm with the extra syrup on the side.