Step 1

Whisk yeast and ¼ cup warm water (110°-115°) in a small bowl; let stand 5 minutes.

Step 2

Heat milk in a small saucepan over medium until just warm. Combine shortening, sugar, and kosher salt in a large bowl. Add warm milk; whisk to blend, breaking up shortening into small clumps (it may not melt completely). Whisk in yeast mixture and egg. Add 3½ cups flour; stir vigorously with a wooden spoon until dough forms. Knead dough with lightly floured hands on a lightly floured surface until smooth, 4–5 minutes. Transfer to a lightly oiled bowl; turn to coat. Cover loosely with plastic wrap. Let stand at room temperature until doubled, about 1½ hours.

Step 3

Preheat oven to 350°. Melt butter in a small saucepan. Lightly brush baking dish with some melted butter. Punch down dough; divide into 4 equal pieces. Working with 1 piece at a time, roll out on a lightly floured surface into a 12x6" rectangle.

Step 4

Cut lengthwise into three 2"-wide strips; cut each crosswise into three 4x2" rectangles. Brush half of each (about 2x2") with melted butter; fold unbuttered side over, allowing a ¼-inch overhang. Place flat in 1 corner of dish, folded edge against short side of dish. Add remaining rolls, shingling to form 1 long row. Repeat with remaining dough for 4 rows. Brush with melted butter, loosely cover with plastic, and chill at least 30 minutes or up to 6 hours.

Step 5

Bake rolls until golden and puffed, 25–35 minutes. Brush with butter; sprinkle sea salt over. Serve warm.