Roast the portobello mushrooms: Slice the portobello mushrooms. In a bowl, toss mushrooms with enough olive oil to lightly coat and sprinkle with kosher salt and fresh ground pepper. Spread the mushrooms in a single layer on a baking sheet and roast for 15 minutes until golden brown, turning once.
Make the dressing: Meanwhile, mince the garlic. In a small bowl, whisk together the garlic, Dijon mustard, red wine vinegar, olive oil, ¼ teaspoon kosher salt, and ⅛ teaspoon black pepper. (Vinaigrette can be stored in the refrigerator for up to 2 weeks; make sure to bring to room temperature and shake well before serving.)
To serve, place the greens on a plate and top with warm mushrooms, cheese shavings, and vinaigrette.