Combine milk and yeast in large bowl; let stand until yeast dissolves, about 5 minutes. Whisk in eggs and melted butter. Add 3 cups flour and stir to form moist sticky dough.
Knead dough on floured work surface, incorporating flour as necessary, until dough is smooth and elastic, and no longer sticky, about 10 minutes. Form into ball. Lightly oil same large bowl; place dough in bowl. Cover with kitchen towel or plastic wrap and let dough rise until doubled in bulk, about 40 minutes.
Heat shortening in deep fryer to 325°F.
Transfer dough to floured work surface. Punch dough down; cover with kitchen towel and let stand for about 15 minutes. Sprinkle dough with flour. Using rolling pin, roll out dough into 12-inch square. Using pizza cutter, trim edges slightly to form perfect square. Cut dough crosswise into 4 equal strips. Cut each strip into 6 pieces, forming 24 rectangles total.
Working in batches, fry dough rectangles in hot oil until golden brown, about 1 1/2 minutes per side. Using slotted metal spoon, transfer beignets to paper towel-lined baking sheet to drain.