Slice the butter into 1/2-inch-thick pieces and place in a single layer on a plate on the counter. Let hang out on the counter until the butter has warmed up to between 63 to 65°F — 35 to 40 minutes. If the butter gets over 65°F, return temporarily to the refrigerator.
While the butter warms, place a folded kitchen towel in the bottom of a wide pot like a Dutch oven. Add 2 inches of water to the pot and bring to a bare simmer, not over 190°F. Whisk the eggs, egg whites, sugar, and corn syrup together in your stand mixer’s work bowl. Place the bowl in the water, atop the folded towel, and continue whisking until the sugar dissolves and the mixture reaches 160°F — 5 to 7 minutes.
Move the bowl to the stand mixer fitted with the whisk attachment and beat on high speed until the mixture thickens and cools to around 75°F — 10 to 12 minutes. Decrease the speed to medium and add the butter, one piece at a time, making sure each piece is fully incorporated before adding the next — 6 minutes. (The mixture may appear to curdle but keep adding in the butter and it will come together.) Stop the mixer and scrape down the sides and bottom of the bowl. Add the salt, boost the speed to high, and beat until the buttercream is smooth and creamy — 2 minutes.
Use immediately or store, tightly covered, in the refrigerator for up to 3 days. If chilled, bring back to room temperature and re-whip with the stand mixer until fluffy — 1 to 2 minutes — before using.