Mix the yeast, water, and sugar in a mixer bowl fitted with a dough hook. Let stand until foamy, about 5 minutes. Add the salt, lemon zest, nutmeg, egg, milk, and 1 1/2 cups flour; mix on medium speed until combined. Add the butter; mix until incorporated. Add 1 3/4 cups flour, and mix until the dough comes together. Turn the dough onto a lightly floured surface. Knead in the remaining 1/4 cup flour by hand until the dough is smooth, about 5 minutes.
Place the dough in a lightly oiled bowl, cover loosely with plastic wrap, and let stand in a warm, draft-free spot until doubled in size, about 1 hour. Remove the dough from the bowl and punch down. Roll out the dough on a lightly floured surface to 1/2-inch thickness. Cut out circles with a 1-inch round cutter; place on a floured baking sheet. Let rise in a warm, draft-free spot for 30 minutes.
Heat the oil in a medium pot or deep fryer until it registers 350 degrees F on a deep fry thermometer. Fry the beignets in batches, rolling them around constantly with a slotted spoon, until golden brown all over, 1 to 2 minutes. Transfer the beignets with the slotted spoon to a platter lined with paper towels and dust with confectioners' sugar. Serve warm with the jam.
Put the blackberries in a food processor and process until coarsely chopped. Transfer the mixture to a large skillet and stir in the sugar and lemon juice. Bring to a boil over medium-high heat, stirring occasionally, and cook until the jam is thickened about 10 minutes. Transfer to a bowl and let come to room temperature.