Bring a large pot of salted water to a boil. Have a bowl of ice water ready nearby.
Meanwhile, in a large bowl, whisk the olive oil with the red-wine vinegar and lemon juice. Stir in the almonds, oregano and red-pepper flakes; season with kosher salt and black pepper.
Working in batches, blanch the asparagus and sugar snap peas until bright green and slightly tender but still crunchy, about 30 seconds. Transfer each batch to the ice-water bath to cool, about 1 minute. Dry the vegetables on paper towels.
Add the asparagus, sugar snap peas, peaches, plums, cherries and mint leaves to the bowl of salad dressing and toss to coat.
Arrange the salad on a large serving platter and nestle in the burrata balls.
