Heat the olive oil in a medium skillet over medium heat. Add the almonds and cook, stirring often, until golden brown, 3 to 4 minutes. Transfer to a paper-towel-lined plate with a slotted spoon.
Increase the heat to medium high and add the grapes, 1/4 teaspoon salt and a few grinds of pepper. Cook, stirring occasionally, until the grapes are tender and blistered in spots, about 3 minutes. Transfer to a bowl with a slotted spoon.
Pour the oil from the skillet into a large measuring cup and whisk in the lemon juice and Dijon. Whisk in the sparkling wine, chives, 1/4 teaspoon salt and pepper to taste.
Add the arugula to a large bowl. Add the grapes on top. Pour the dressing over the salad. Top with the almonds, grate over some fresh Parmesan and serve immediately.