For dukkah: Place pistachios and walnuts in a skillet and place over medium-low heat. Lightly toast nuts for 4 to 5 minutes, shaking the pan frequently to prevent burning and lowering the stove temperature as needed.
Remove from heat and transfer to a food processor. Pulse the nuts until evenly ground (I like my dukkah to be on the finer side, but you can make the nuts coarser).
Transfer ground nuts to a mixing bowl.
Pour sesame seeds into the skillet and return skillet to heat. Toast for 2 to 3 minutes and transfer sesame seeds to the nut mixture.
Pour coriander, cumin seeds and peppercorns in the skillet and toast for 3 to 4 minutes or until fragrant.
Transfer toasted seeds to the food processor (or a spice grinder) and finely grind mixture. Transfer spices to bowl with nuts and sesame seeds, add sea salt flakes and stir until completely incorporated.
Store dukkah in a cool, dry place, in an airtight container for up to 3 months.
For vinaigrette: Place all ingredients, except oil, into a large bowl and whisk together. While whisking, slowly drizzle oil into the mixture until fully incorporated. Season with salt and pepper.
To assemble: Top vinaigrette with toasted pita and cucumbers and toss together. Refrigerate for at least 30 minutes and up to 2 hours. Before serving, add peaches to the salad and gently toss together. Pour salad onto a platter and top with burrata. Sprinkle with a generous amount of dukkah and sone chives and serve.