Step 1Fill a large saucepan with water, salt the water to taste like the ocean, adding approximately 1 tablespoon of salt to each quart of water, and bring it to a boil over high heat. Fill a bowl with ice water and line a plate or small bowl with paper towels. Place a wire strainer in the sink. Add the English peas to the boiling water and cook them until they turn bright green but are still crunchy, about 30 seconds. Quickly drain the peas into the wire strainer and plunge them, still in the strainer, in the ice water to cool completely, about 1 minute. Transfer them to the paper towels to drain.
Step 2Pull the strings off the sugar snap peas, discard and slice the sugar snap peas one-eighth-inch thin on such an extreme bias that you are almost slicing them lengthwise. Put the peas, sugar snap peas and mint in a medium bowl. Sprinkle with the grated Parmigiano-Reggiano, season with salt and pepper, drizzle with the olive oil, and toss to combine the ingredients and coat the vegetables with the seasonings.
Step 3Drape four slices of speck in a rosette-like pattern on each of four salad plates, dropping the slices onto the plate so they stand up slightly rather than placing the slices flat against the plate. Cut the burrata into four equal segments and nestle one segment in the center of each "rosette." Pile the dressed peas on top of the burrata, allowing a bit to fall onto the speck below, and use a microplane or other fine grater to grate a light layer of Parmigiano-Reggiano over each plate, and serve.