Preheat oven to 400°F. Spread on a foil-lined baking sheet in a single layer, toss with 1 tbsp olive oil and a 1/4 tsp salt, and roast until tender and slightly caramelized at the edges, about 25–30 minutes, flipping halfway through.
Toast the hazelnuts:
If using raw hazelnuts, spread 1/3 cups hazelnuts, raw or roasted on a dry skillet over medium heat and toast, stirring frequently, until fragrant and lightly golden, about 4min. Let cool and roughly chop.
Make the dressing:
In a small bowl or jar, whisk together 3 tablespoons olive oil, 1 ½ tablespoons red wine vinegar, 1 teaspoons Dijon mustard, 1 teaspoons honey, ¼ teaspoons salt, and ⅛ teaspoons black pepper until emulsified. Taste and adjust seasoning.
Assemble the salad:
Arrange 3 cups mâche and 2 cups arugula on a large platter or in a wide salad bowl. Top with the roasted beet pieces, then scatter over 3 ounces feta, crumbled and the toasted 1/3 cups hazelnuts.
Dress and serve:
Drizzle the vinaigrette over the salad just before serving. Finish with a sprinkle of 2 tablespoons fresh parsley, chopped. Serve immediately.
Notes
Make-ahead: Beets can be roasted up to 3 days in advance and stored in the fridge. Dress the salad right before serving so the greens stay fresh.