Prepare a large pot of salted water. Wash and rinse the rainbow chard. Tear the large leaves from the stem and chop very coarsely. Boil the greens in the salted water for 3 minutes and strain.
Peel and mince the shallot. Coat a sauté pan with olive oil and warm the pan on medium heat. Add the minced shallots and sauté until translucent. Add the rainbow chard and continue sautéing until well-cooked - about 8 minutes. Season with salt and pepper. Remove from heat.
Slice the bread into six 2-inch-thick slices. Grill or toast the bread until dark golden and charred. Slice the garlic cloves and rub the cut side against the toasted bread.
Arrange the crusty bread slices on a plate. Drizzle with extra virgin olive oil. Spoon the sautéed greens onto the bread. Shave the fresh Parmigiano and arrange a few shavings on top. Grate and juice the lemon. Add grated lemon zest. Drizzle with fresh lemon juice and more extra virgin olive oil. Top with a pinch of North Fork Sea Salt. Serve warm.