Step 1

For the Hawaiian rolls: Into the bowl of a stand mixer fitted with the dough hook, add the flour, yeast, kosher salt and turmeric and mix on low to combine. Add the pineapple juice, brown sugar, honey, whole egg and yolks and vanilla. Increase the speed to medium and mix to form a somewhat stiff, shaggy dough, about 1 minute. With the mixer running, add the butter, 1 tablespoon at a time, allowing each to fully incorporate before adding the next. Once all of the butter is added, mix on medium-high to make a smooth, glossy, somewhat-wet dough, 10 to 13 minutes. Oil a large bowl and add the dough, turning to coat. Cover and let rise until doubled in size, 1 1/2 to 2 hours.

Step 2

Punch the dough down and divide into 12 equal pieces. Butter a quarter sheet pan. Form each piece into a ball and roll on the counter encased in your palm to tighten up the ball. Lay the balls in a 3-by-4 grid in the buttered pan. Cover loosely and let rise until puffed but not quite doubled, 1 hour to 1 hour 15 minutes.

Step 3

Arrange an oven rack in the middle position. Preheat the oven to 350 degrees F.

Step 4

When the rolls have risen, beat the reserved egg whites with a splash of water and brush over the rolls. Sprinkle with flaky salt. Bake until puffed and a deep golden brown; start checking at 20 minutes. Cool in the pan on a cooling rack for 10 minutes, then remove to cool completely.

Step 5

Meanwhile, for the sausage, combine the pork, maple syrup, fennel, smoked paprika, salt, sage, thyme, onion powder and nutmeg in a large bowl and mix well with your hands to make a smooth mixture. Form into 12 very thin patties.

Step 6

Heat a large, cast-iron or nonstick pan over medium heat. Add as many patties as will fit. Cook until browned on the bottom, 1 to 2 minutes, then flip and cook, 1 to 2 minutes until browned on the other side, crispy and cooked through. Add a slice of American cheese on top of each patty, to melt. Remove to a cooling rack. Repeat with the remaining sausage patties and cheese.

Step 7

To assemble: Split the rolls and add the cheesy sausage patties in between. Serve right away with maple syrup for dunking, if desired.

Step 8

The rolls and sausage mixture can be made a day ahead. Leftover rolls will keep for several days in an airtight container at room temperature. Leftover cooked sausage patties will keep for 1 to 2 days in the refrigerator.