Whisk together yeast and ¾ cup of the warm milk in the bowl of a stand mixer; let stand until foamy, 5 to 10 minutes.

Add flour, butter, sugar, eggs, salt, and remaining 1 ¼ cups warm milk. Beat on low speed with paddle attachment until just combined and a sticky dough forms, about 1 minute.

Remove paddle attachment and attach dough hook; knead on medium speed until dough is smooth, elastic, and beginning to pull away from sides of bowl, 8 to 10 minutes. If necessary, add 1 tablespoon of additional flour at time, but not exceeding ¼ cup total.

Transfer dough to a large bowl lightly greased with cooking spray, turning dough to coat top.

Cover and let rise in a warm place (75°F to 80°F) until doubled in size, about 1 hour.

Preheat oven to 400°F. Coat 2 (8-inch) square baking dishes with cooking spray. Punch down dough and turn out onto a clean work surface. Divide dough into 18 even portions (about 3 ounces each). Working with 1 dough portion at a time, cup dough with palm and roll against counter into a smooth, tight ball. Repeat with remaining dough portions.

Arrange dough balls evenly between prepared baking dishes; cover and let rise in a warm place (75°F to 80°F) until dough is almost doubled in size, about 30 minutes.

Bake in preheated oven until tops are deeply browned and a thermometer inserted into center of rolls registers between 180°F and 190°F, 16 to 20 minutes. Remove from oven and brush tops with additional butter; let stand about 10 minutes before serving.