Adjust oven rack to center position and preheat oven to 350°F./175°C. Fit a cooling rack onto a rimmed baking sheet and then line it with a sheet of parchment paper (or aluminum foil). Brush the parchment paper with a bit of olive oil.
Slice the mushrooms. If using King Oyster mushrooms, slice lengthwise into 1/8th- to 1/6th-inch slices. If using shiitake mushrooms, slice lengthwise into 1/4-inch thin slices.
Combine the tamari, 1 tablespoon olive oil, sesame oil, liquid smoke, maple syrup, and smoked paprika in a shallow dish and whisk to combine.
Add the sliced mushrooms, and stir gently to combine. Marinate the mushrooms at room temperature for at least 20 minutes, or up to 60 minutes.
Arrange the mushrooms in a single layer on the baking sheet.
Bake the mushrooms for 20 minutes. Flip mushroom slices carefully with a thin spatula and continue baking until they are well-browned and chewy but crisp, about 20 minutes longer.
Remove from the oven and transfer the mushrooms to a paper towel-lined plate to drain any excess oil.