Set a large pot of water over high heat to boil. Once boiling, heavily salt the water and then add in three pounds of Yukon gold potatoes. (If your potatoes are larger than 3 inches long, cut them in half before adding to the boiling water.)
Meanwhile, preheat oven to 425 degrees. Line a large, rimmed baking sheet with parchment paper and set aside.
While the potatoes boil, microwave ¾ cup butter in a glass bowl until melted, about 1 minute. Stir in the roasted garlic, ¼ cup olive oil, ½ cup shredded parmesan, and the leaves of 4 thyme sprigs, mixing well to combine. It will form a sort of paste.
Once potatoes have just become fork tender, after about 20 – 25 minutes, remove from heat and drain.
Allow potatoes to cool for a few minutes, or until they can be reasonably handled.
Place potatoes on prepared baking sheet, about 4 inches apart. Using the bottom of a heavy glass or mason jar, press down on the potatoes so they squish out from under the glass. Take care not to press too hard! We’re just looking to burst the potato, exposing the inside and creating an even layer.
Sprinkle kosher salt and freshly cracked black pepper over the potatoes.
Generously pour the flavored butter mixture over the potatoes, evenly distributing the roasted garlic and parmesan, as it sinks to the bottom of the liquid.
Bake potatoes in preheated oven for 20 minutes, or until deeply golden brown.
Remove and garnish with extra parmesan cheese, flaky salt, and red pepper flakes, if desired.