For the Pesto: In a food processor, combine basil, arugula, garlic, almonds, lemon juice, and lemon zest. Pulse until almonds are broken up into fine pieces. Add olive oil and process until fairly smooth. Season with salt. Transfer to an airtight container and refrigerate until ready to use. The pesto can be made in advance and refrigerated for up to 3 days.
For the Toasts: Using the largest holes on a box grater, shred tomatoes into a bowl. (Hold the tomatoes skin side out and discard the skins after grating.) Stir in olive oil. Season with salt and black pepper or red pepper flakes. Let stand for 10 minutes.
Lightly brush sourdough on each side with olive oil and toast on both sides until lightly browned. (You can do this in a toaster, in an oven, under the broiler, or on a grill.)
Preheat broiler and set top oven rack about 6 inches from broiler element. Using a fine-mesh strainer, strain tomatoes of their excess liquid and reserve the liquid for another use (it's delicious as part of a vinaigrette for a simple salad).
Spread strained tomatoes in an even layer on the toasts. Broil to dry and warm the tomatoes, stopping before the edges of the bread burn, about 1 minute. Cut each toast into 4 even pieces.
Cut burrata into small portions and dollop on top of toasts. Spoon some of the pesto on top. Serve.
