Position a rack in the lower third of the oven and preheat to 425 degrees . In a large skillet, heat 2 tablespoons olive oil over medium-high heat. Add the scallion whites and cook until softened, about 1 minute. Add the ham and cook, stirring occasionally, until heated through, about 3 minutes. Transfer to a medium bowl.
Return the skillet to medium heat and add the remaining 1 tablespoon olive oil. Add the mushrooms and cook, stirring occasionally, until their liquid is cooked off, 6 to 8 minutes. Stir into the ham mixture along with the cream cheese and scallion greens. Let cool.
Transfer 1 tablespoon beaten egg to a small bowl and stir in 1 tablespoon water. Stir the mustard into the remaining beaten eggs and season with a pinch salt.
Center a piecrust round over a 9-inch pie plate, pressing the dough into the sides and bottom. Fill with the ham mixture, spreading it evenly. Pour in the mustard-egg mixture, tilting the plate to distribute evenly. Cover completely with the tomato halves, cut side down.
Place the remaining piecrust round on a lightly floured work surface. Cut into 3/4-inch strips. Place the longest strips over the center of the pie, forming a cross. Lay the remaining strips parallel to the first strips, alternating directions and spacing them evenly apart, to form a lattice top. Trim off any excess ends. Roll the rim of the bottom crust up over the lattice and crimp. Brush the top with the egg wash.
Transfer the pie to a baking sheet and bake until the egg mixture is set, about 50 minutes. Let cool for 30 minutes before serving.