Preheat oven to 400 F.
Pierce whole russet potatoes with a fork (this allows steam to escape and prevents them from bursting). Arrange on baking sheet and roast whole potatoes for 1 hr - 1 hr 15 mins, or until completely cooked through and fork tender.
When cooked potatoes have slightly cooled, peel skin off or slice potatoes in half lengthwise and scoop out potato in large bowl. Discard skins.
In large bowl, mash potatoes with a potato ricer, food mill, handheld masher, or a fork, making sure there are no large clumps. You want it to be tender and light.
Add 75% of the flour, and the entire lightly beaten egg, the salt, and gently mix by hand to combine.
If the dough is still very wet, add the remaining flour and combine with hands, scraping down excess flour from sides of bowl as needed to incorporate. The dough should easily form a ball and be slightly sticky, but won’t stick to clean, dry hands. Add more flour if needed, one tablespoon at a time.
Transfer dough to a lightly floured, flat work surface, like your kitchen counter or table.
Cut the dough into four equal pieces.
Using the palms of your hands, gently roll each piece into a thin rope shape, about ½" in diameter, or about 20 inches long. Cut each rope of dough into 1" pieces, and transfer gnocchi to a floured surface, like a baking sheet with lightly floured parchment paper.
Once you've portioned out the gnocchi pieces, you'll make the browned butter, then cook the gnocchi, and transfer cooked gnocchi into the same skillet as the browned butter sauce to heat and toss right before serving.