How to Make It
Combine oil, basil, and chives in a food processor. Process until very smooth and fully incorporated, 1 to 2 minutes, stopping to scrape down sides of bowl as needed. Pour mixture through a fine-mesh strainer, pressing on solids to extract oil. Discard solids.
Arrange prosciutto on a serving platter. Gently toss arugula with 2 tablespoons herb oil in a large bowl. Add peaches and gently toss to combine. Arrange mixture on top of prosciutto. Tear burrata into pieces and place on top of salad. Drizzle with 2 more tablespoons herb oil. (Reserve remaining herb oil for another use.)
Top with chopped almonds and season with salt and pepper.