Melt 1 stick of the butter in a small saucepan, then bring to a boil and cook until there are light brown specks in the butter, 3 to 4 minutes. Pour the browned butter into a small bowl and whisk in the sugar until dissolved. Whisk in the milk; the mixture should be lukewarm, around 110 degrees F. Next, whisk in the active dry yeast. Set aside until foamy, 5 minutes.
In a large bowl, mix together the flour and salt. Make a well in the center, add the eggs and whisk the eggs together with a fork. Pour the browned butter mixture into the well. Using a fork, gradually incorporate the flour into the liquids. Continue stirring just until a shaggy dough comes together. Transfer to a lightly floured work surface and knead with floured hands until the dough becomes very smooth, about 5 minutes.
Lightly grease a large bowl with olive oil. Add the dough, turning it once to coat. Cover with a clean kitchen towel and let rise until doubled in size, about 1 hour.
Punch down the dough. Melt the remaining 1/2 stick of butter and use half of that to grease the bottom and sides of a 9-by-13-inch baking dish. Divide the dough into thirds and divide each third into 10 balls. To create smooth tops: roll each ball between your palms, and then stretch the dough across the top of the ball, pinching the excess at the bottom.
Place the dough balls, seam-side down, in the prepared baking dish -- 5 to the short side, 6 to long side. Cover and let rise until slightly puffy, 30 minutes. Brush the tops with the remaining butter and sprinkle generously with flaky sea salt.
Bake until the tops are golden brown and the rolls sound hollow when lightly tapped, 15 to 17 minutes. Serve warm.