To make the mini pork sausages: In a medium bowl, place the pork; season with salt and pepper. Cover and refrigerate for at least 1 hour or up to overnight. Place your meat grinder in the fridge to chill too.
Remove and combine the meat with the rice and ice. Pass the mixture through the grinder's finest setting, adding the rice and ice as you go. Then grind the mixture a second time, working quickly to keep the mixture as cold as possible.
Transfer the mixture to a piping bag and cut a 3⁄4-inch opening at the tip. Lay an 18-inch piece of plastic wrap on the work surface, and starting at one end, pipe a tube of sausage mixture in a straight line along the edge. Roll up the plastic tightly to create a long sausage shape. Using butcher's twine, tie off one end. Then pinch and twist the sausage every 2 inches and tie with more twine to form small links. Repeat with an additional piece of plastic wrap and the remaining filling. Chill until ready to cook, up to 1 day.
Heat a large pot of water until warm but not simmering (about 145°). Add the wrapped sausages and poach until a meat thermometer reads an internal temperature of 145°, about 10 minutes. Transfer to a bowl filled with ice water. Unwrap the sausages and pat dry.
Make the dough: IN a large bowl, sift the flour, cream of tartar, baking soda, and salt. Add the butter and shortening, and using a pastry cutter or your fingers, cut in the fat until only pea-size crumbs remain. Add the buttermilk and mix with a fork, just until a shaggy dough forms.
Turn out the dough onto a large sheet of plastic wrap. Gently form into a 7x9-inch rectangle with even thickness. Wrap in plastic, then chill overnight.
Remove the dough onto a floured work surface. Roll it out to 1⁄2-inch thickness, maintaining the rectangular shape. Fold the rectangle into thirds like you would a business letter. Flour your work surface again; turn the folded dough over so the seam is on the counter and the short end is facing you. Roll it out again to a 1⁄2-inch-thick rectangle. Fold and repeat 2 more times (if dough is too elastic to roll out, refrigerate 30 minutes after each time you fold). Wrap in plastic and refrigerate for at least 1 hour or up to overnight.
Assemble the pigs: Preheat the oven to 425° and set a rack in the center of the oven. Line a baking sheet with parchment paper. In a small bowl, beat the half & half and egg yolk together until combined.
Unwrap the dough and cut into 2 equal pieces. On a lightly floured work surface, roll out the first half into a 9x13-inch rectangle, slightly thinner than 1⁄8 inch (keep the remaining dough refrigerated). Cut the dough into 4 strips approximately 2 1⁄4 inches wide. Working with 1 strip at a time, brush about 2 inches of one end lightly with the egg mixture. Smear a mini pork sausage with 1⁄4 teaspoon mustard and place it at the prepared edge. Roll tightly until the sausage is wrapped completely with dough and there is a little overhang from the seam. Cut the wrapped sausage off the strip, roll the seam to the bottom so it is against the counter, and press down gently to seal. Transfer to the prepared baking sheet. Repeat with the remaining dough and sausages (you should be able to wrap 3 sausages with each strip of dough), wrapping and chilling the pigs in a blanket until you are ready to bake. Reserve any remaining egg wash.
Brush the pigs in a blanket with the egg wash and transfer to the oven. Bake until the dough has puffed and browned, about 22 minutes. Let cool for 5 minutes before serving.