Bring a medium pot with 1 cup water to a boil for shiitakes. Thoroughly rinse produce and pat dry. Cut cremini mushrooms into 1/4" slices. Dice roasted red pepper into 1/2" pieces. Coarsely crush or chop hazelnuts. Trim and peel both shallot and carrot, then cut to 1/4" small dice. Place dried shiitake mushrooms in a small bowl and pour 1 cup boiling water over them. Cover with a plate or plastic wrap and allow to sit for 15 minutes.

Return medium pot to stove over high heat. Add 1 Tbsp. olive oil into hot pot. Add cremini mushrooms and cook until lightly browned, about 3 minutes. Reduce heat to medium-high and add carrots, shallots, and a pinch of salt and pepper. Cook 2 minutes, add barley and 2 cups water. Bring to a boil, reduce to a simmer, cover, and cook about 25 minutes, or until liquid has been absorbed and barley and carrots are tender. Season with salt and pepper to taste. Set aside.

While barley is cooking, place a small pan over medium-high heat. Place crushed hazelnuts in pan and toast 2 minutes while constantly stirring. Remove hazelnuts from pan and set aside to cool. Lightly brush out pan and return to stove.

Check bowl with shiitakes for grit or silt in the bottom. If it's gritty, strain liquid through a wire-mesh strainer and reserve. Pour shiitakes and their liquid into small pan, along with sherry vinegar and soy sauce. Place over high heat and bring to a boil, reduce heat to medium, and cook until liquid has completely reduced. The shiitake soaking liquid is full of flavor, and reducing it concentrates it further. Flip shiitakes often and be careful not to burn them.

Cut pickled shiitakes into 1/4" slices and toss together with arugula and roasted red peppers in a mixing bowl.

Add a serving of mushroom barley to a plate or shallow bowl. Place arugula shiitake salad on top and garnish with crushed hazelnuts and shaved Parmesan cheese.