Combine the milk, brown sugar, and salt in the pot of electric pressure cooker. Turn on the sauté setting. Warm the milk and sugar until the sugar dissolves, stirring constantly with a wooden spoon, about 3 minutes. Add the rice and salt and stir to combine
Close and lock the lid. Pressure-cook for 10 minutes at HIGH pressure.
When the cooking cycle has completed, turn off the pressure cooker. Do not allow it to switch to the "keep warm" setting, as this can cause the bottom to scorch. Allow the pressure to release naturally, this takes 15 to 20 minutes. Unlock the lid and open the lid away from your face.
Whisk the pumpkin, egg, pumpkin pie spice, and vanilla together in a small bowl. (Doing this prevents the egg from scrambling when it hits the hot pudding. So it's not a step to skip!)
Whisk the rice pudding to combine. Add the pumpkin mixture. Stir to combine.
Turn on the sauté setting. Cook the pudding, stirring constantly, until it thickens, about 3 minutes. Remove the pot from the electric pressure cooker and pour the pudding into a greased 9 x13-inch pan. Serve warm or press a piece of plastic wrap onto the surface of the pudding. (This prevents a skin from forming.)
Store pudding in the refrigerator for up to four days.