Scrub the potatoes well, pat dry and then lightly coat with olive oil and a generous sprinkling of salt and pepper. Roast potatoes in a preheated 375 degree F oven for 15 to 20 minutes or until just cooked through. Cool and quarter lengthwise.
Remove stem and scrape black gills from the portobellos. Brush with olive oil and season with salt and pepper. Grill, pan grill or roast until just cooked through. Cool and slice each mushroom into 12 thick slices.
To construct a stack lightly oil a baking sheet and arrange 4 quarters of potato interspersed with 3 slices of mushroom side by side in the same direction. Top with a half ounce or so of cheese and then arrange another layer at right angles to the first. Finish with a top layer of potatoes and a dollop of cheese. Construct 3 more stacks in the same manner.
When ready to serve, place baking sheet in a preheated 400 degree F oven for 6 to 8 minutes to heat vegetables through and barely melt the cheese. While vegetables are heating toss the greens with vinaigrette. Place a stack on a heated plate and top with a quarter of the dressed greens. Scatter the diced peppers around and drizzle the plate with the basil oil. Serve immediately.
Whisk all ingredients until sugar is dissolved. Store covered in refrigerator up to 1 week.