For the salsa verde: In the bowl of a food processor, pulse together the kale and garlic until roughly chopped. Add the capers, vinegar, Dijon mustard, anchovy paste and 3/8 teaspoon salt and pulse 1 or 2 times more. Add the olive oil and run the processor for 5 seconds. Set aside to let the flavors marry.
To assemble the bruschetta: Preheat a grill pan over high heat. Drizzle both halves of the bread evenly with the olive oil. Grill on each side until golden brown and charred in a few spots, about 2 minutes per side. Remove the bread to a cutting board. Spoon the salsa verde evenly over each half of the bread. Tear the burrata over the salsa verde and sprinkle the pine nuts and tomatoes evenly over the tops. Cut each bread half into slices and serve.