In advance: Slice the tip off of a head of garlic and wrap it in foil. Drizzle with a small amount of olive oil and a pinch of salt. Wrap tightly and roast the garlic at 350 degrees F. for 35-40 minutes. Remove and let cool.
1) Add warm water (not hotter than 110 degrees F.) to the bowl of your KitchenAid® Artisan® Mini Mixer. Stir in yeast, brown sugar, and salt. Let sit for 5 minutes until yeast is bubbling.
2) Stir in melted butter and bread flour in 1 cup batches until the pretzel dough comes together. Attach the dough hook attachment and knead the dough for 6-7 minutes on medium (setting 4) speed. If dough is very sticky and sticking to the bottom, add more bread flour by the ½ cup. Final dough should be soft, but not sticky to the touch.
3) Turn the mixer speed down to low and add the cheddar cheese and mashed roasted garlic. Mix to combine. Remove the dough from the mixing bowl and knead a few times with your hands to make sure ingredients are distributed evenly. Wipe bowl clean and drizzle with neutral oil. Return dough to bowl and cover loosely. Let rise for one hour.
4) Preheat oven to 450 degrees F. Scoop dough out of mixing bowl and roll out on a clean surface dusted with flour. Cut dough into strips about 6-8 inches long. You should get about 20 strips out of the dough.
5) To make a pretzel knot, roll a dough strip under your hands until it’s about 8-10 inches long. Then fold it in half and twist the dough a few times. Tuck the ends of the dough under itself and through the hole near the top of the dough, forming a knot. Repeat with all the dough strips.
6) Bring 8 cups of water to a simmer. Slowly stir in baking soda. Working in batches, boil the pretzels for about 20 seconds per side in the mixture. Remove them and let them drain on a few paper towels. Place boiled pretzels on a baking sheet lined with parchment paper.
7) Brush pretzels with egg wash (1 egg whisked with 1 tablespoon water) and sprinkle lightly with coarse pretzel salt.
8) Bake pretzel knots at 450 degrees F. for 12-15 minutes until they are browned nicely and crunchy on top.
Let the pretzels cool briefly before serving. They are great at room temperature also. Serve with whole grain mustard. Store any leftover pretzel knots at room temperature in an air-tight container for up to 3 days or freeze them for longer storage.