Heat a non-stick skillet, brushed with a light coating of olive oil over medium-high heat.
While pan heats, carefully slice the mushroom caps into 1/8" slices.
Brush each side of the mushroom slices with olive oil and sprinkle with salt and pepper. Add slices, in a single layer to the hot skillet and cook 1-2 minutes per side, until edges are browned and middle is cooked through. Make sure not to crowd the pan, so that the mushrooms will brown and not steam. This may need to be done in batches, depending on the size of your pan.
Remove cooked mushrooms to a paper towel lined plate and place in the refrigerator to cool for thirty minutes.
While mushrooms are cooling, combine all of the dressing ingredients in a small mixing bowl and whisk to incorporate. You will have more dressing than you will need for this salad. Leftover dressing may be kept refrigerated for up to 3 days.
Assemble the salad by drizzling 1 tablespoon of dressing over a salad plate. Shingle the cooled mushrooms in a circular pattern in the center of the plate. Top mushrooms with one cup of arugula and sprinkle with shaved cheese and chopped chives. Arrange quartered red and yellow tomatoes around plate. Repeat on a separate plate with remaining ingredients. Serve additional dressing on the side, if desired.
