Combine the arugula, parsley, cheese, pine nuts, shallot and some salt and pepper in a food processor, and process until coarsely chopped. With the motor running, pour in the olive oil and blend until emulsified. The mixture should be slightly loose; if it is stiff, add a splash of water and process briefly to combine.
Preheat the grill to high heat. Brush the cut sides of the peaches with canola oil and season with salt and pepper. Grill cut-side down, covered, until charred, about 3 minutes. Flip over and continue grilling, covered, until soft, 3 or 4 minutes longer. Arrange the peaches cut-side up on a serving platter and drizzle with some olive oil. Spoon some pesto into the center of each peach half, then fold a piece of prosciutto and place it on the edge. Serve immediately.
Refrigerate leftover pesto, covered, for 3 days.