Stir the yeast into ⅔ cup of the warm water and let it stand until foamy, 5 to 10 minutes.
In a large bowl, mix the warm mashed potatoes, sugar, shortening, eggs, salt and the remaining ⅔ cup warm water.
Stir in the yeast mixture. Add the flour gradually until a soft, slightly tacky dough forms, then knead until smooth.
Place the dough in a greased bowl, cover with a kitchen towel and let it rise until doubled, about 1 hour.
Punch down the dough and shape it into 1- to 2-inch balls. Arrange them close together in a greased pan that fits your air fryer basket.
Cover and let rise until doubled, 30 to 50 minutes.
Air fry at 320°F for 15 to 20 minutes, until deep golden brown and 190°F in the center. Repeat with the remaining rolls.
Brush with melted butter, sprinkle with fresh herbs and serve warm.