For the potato bagels: Place the chopped potatoes in a large pot and cover with water. Bring to a simmer and cook until a fork pokes easily into the potatoes; begin checking for doneness at about 10 minutes. Drain, reserving 1 cup of the cooking water in a medium bowl. Rice the potatoes and set aside.
Add 1/2 cup cool water to the bowl with the hot cooking water; the temperature of the water should even out to warm (105 to 110 degrees F). If it's too cold, microwave it a little; if it's too hot, let it sit for a few minutes to cool. Add the yeast and 1 teaspoon of the brown sugar, give it a little swirl and let it sit until foamy on top, about 5 minutes.
In a large bowl or the bowl of a stand mixer fitted with a dough hook, combine the flour, salt and remaining 2 tablespoons brown sugar. Add the mashed potato, yeast mixture and 4 1/2 teaspoons of the barley malt and mix to form a stiff dough. Knead in the stand mixer until smooth and slightly sticky, adding additional flour if the dough is sticking to the bowl or too sticky to handle, about 5 minutes (or turn out onto a floured surface and knead by hand for 7 to 10 minutes). Transfer to a large oiled bowl, cover with plastic wrap and refrigerate overnight.
When ready to bake, let the dough sit at room temperature for 1 hour.
Turn the dough out onto a clean work surface and divide into 12 or 16 equal parts, depending on whether you want larger or smaller bagels. Stretch each part into a smooth ball, making sure to seal any dough seams well. Shape the bagels by sticking your thumb through the center of each ball and using your fingers to gently stretch a 2-inch hole. Cover the bagels with a towel and let rise for 15 minutes.
Meanwhile, preheat the oven to 450 degrees F; line 2 baking sheets with parchment paper and grease them well with cooking spray.
Bring a large pot of water to a boil; add the baking soda and remaining 3 teaspoons barley malt. Working with 3 or 4 bagels at a time so as not to crowd the pot, boil them for 1 minute on each side (use a timer for this). With a slotted spoon or spatula, lift them out of the water, allowing excess water to drip off, and then transfer them to the prepared baking sheets, spacing them about 2 inches apart. (If making 16 bagels, you will likely need to bake a third pan of bagels, which you can do after the first 2 pans come out of the oven.)
With a serrated knife, make 3 cuts in a triangular shape on top of the bagels and dust them lightly with flour. Bake the bagels for 10 minutes, then switch racks, rotate the baking sheets 180 degrees and bake until golden brown, 5 to 8 more minutes. Let cool slightly.
Set aside 9 of the bagels for making the breakfast sandwiches. Cool the remaining bagels fully, then transfer to plastic bags and freeze (slice them before freezing if you'd like, so you can put them directly from the freezer into the toaster). Reheat in the toaster.
For the baked eggs: Preheat the oven to 400 degrees F. Grease a 9-inch square baking dish with cooking spray and line with parchment paper so that wings come up 1 inch off 2 sides.
Sprinkle the hash browns in an even layer on the bottom of the prepared baking dish. Sprinkle the Swiss cheese in an even layer on top of the hash browns. Whisk together the milk, paprika, salt, eggs, some black pepper and a few shakes of hot sauce if using in a medium bowl. Gently pour the egg mixture over the cheese and hash browns.
Bake until the eggs are set, about 20 minutes. Allow to cool for a few minutes to allow the eggs to set further. Slice into 9 squares.
For the sandwiches: Slice the reserved 9 bagels in half. Sandwich one egg square between each bagel with a slice each of cheese and Canadian bacon. If not eating right away, wrap in parchment paper and freeze for up to 3 months. Reheat in the microwave for 1 minute, flip and heat for an additional 1 to 2 minutes until heated through.