Preheat oven to 400°F.
Lightly toast sourdough slices. Lightly brush each slice with olive oil. Allow to cool, then place toasts on a parchment lined baking sheet.
To shave Brussels sprouts, wash and pat dry. Trim the rough, woody stems and remove outer leaves. Using a mandoline or sharp knife, thinly slice each bulb in from top to stem.
In a large bowl, toss shaved sprouts with 2 Tbsp olive oil, salt & pepper. Spread on another parchment lined baking sheet and bake for 15 minutes. Remove from oven, toss with parmesan and balsamic vinegar, then return to oven and bake for another 15 minutes, until browned and slightly crispy.
Top each toast slice with a mound of Brussels sprouts mixture. In a small bowl, break burrata and spoon generously over sprouts. Heat in oven until cheese is melted 2-3 minutes. Remove from oven, drizzle with honey and sprinkle crushed pistachios over the top.