Season the mushroom with 1 teaspoon salt and set aside in a bowl in the refrigerator to allow it to release any moisture. Put the Cheddar and pork on 2 quarter-sheet trays and freeze just until firm, about 1 hour.
Using a stand mixer fitted with the coarse grind attachment, grind the pork into the bowl of the stand mixer. Add 1 tablespoon salt, the pepper and sage and toss to combine. If the meat feels warm, place it in the freezer for a few minutes until it cools. Switch to the fine grind attachment and regrind the meat mixture. Add the milk and mushrooms to the meat mixture and switch to the paddle attachment. Mix until evenly dispersed. Add the Cheddar and stir until combined.
Preheat the oven to 450 degrees F. Line a baking sheet with a wire rack.
Bring a 6-quart saucepan full of water to a boil. Divide the meat mixture into 8 equal portions or use a 1-cup ice cream scoop to measure out eight portions. Using plastic wrap as the casing, roll each portion of meat into a 6- to 7-inch sausage link. Twist the ends tightly and wrap a second piece of plastic wrap around each sausage to prevent water from getting in. Place the sausages in the boiling water and cook until the outsides turn whitish and set up, about 6 minutes. Remove from the water, take off the plastic wrap and place on the prepared baking sheet. Bake the sausages until golden brown and firm to the touch, 15 to 20 minutes.
Place on a bun, top with mustard, and enjoy.
Grinding your own pork for this recipe is worth it because you can control the amount of fat going into the sausage but purchased ground pork will work here, too.