In a large bowl, sprinkle yeast over 1/4 cup cool (70°) water. Let stand until dissolved, 5 minutes. Stir in sugar, table salt, herbs, butter, whole egg, and milk; add 3 1/4 cups flour and stir to moisten.
Knead dough on a lightly floured board until elastic and not sticky, 15 minutes; add flour as needed to prevent sticking. Shape into 12 equal balls; place in a well-buttered 9- by 13-in. pan.
Let dough rise in a warm place until doubled, 45 to 60 minutes. Meanwhile, preheat oven to 350°. Brush rolls with beaten egg and sprinkle with coarse salt. Bake until deep golden, 25 to 30 minutes.
Make ahead: Prepare through step 2; immediately wrap airtight (do not let rise) and chill up to 1 day. Proceed with step
Note: Nutritional analysis is per roll.