Preheat the oven to 400 degrees F. Heat the rice as the label directs. Combine in a large bowl with the scallion whites, tomato, andouille and cheese. Season with salt and pepper and mix well.
Separate the biscuit dough into 8 equal pieces. Dust with flour and flatten each into a 6-inch round with a rolling pin; brush off any excess flour. Add about 1/3 cup of the rice mixture to the center of each, then fold the dough over the filling and pinch the edges together to seal. Cut three slits in the top of each pie with a knife. Transfer to a rimmed baking sheet and brush with the beaten egg. Bake until the dough is puffed and golden and the filling is hot, about 15 minutes.
Meanwhile, halve the cucumber lengthwise, then thinly slice crosswise. Combine the cucumber, sugar and a few pinches of salt in a large bowl. Rub the seasonings into the cucumber slices, then let stand 5 minutes. Add the vinegar, scallion greens and parsley and toss to combine. Serve with the hand pies.