Aged Rib Eye with Onion Purée
['6 tablespoons unsalted butter, divided', '2 medium Vidalia or Maui onions (about 1 1/4 pounds total), very thinly sliced on a mandoline (about 3 packed cups)', '1/4 cup low-salt chicken stock', '1 tablespoon buttermilk', 'Kosher salt', '1 tablespoon vegetable oil', '1 28-ounces dry-aged rib-eye steak (about 2" thick), standing at room temperature for 1 hour', 'Coarse sea salt', 'Assorted soft herb sprigs (such as tarragon', 'flat-leaf parsley', 'and chervil)']
Heat 2 tablespoons butter in a 12" heavy skillet over medium-low heat. Add onions and cook, stirring constantly, until translucent (do not brown), 10-12 minutes. Add stock and 3/4 cup water. Bring to a simmer, reduce heat to low, cover, and continue to simmer until onions are falling apart, about 20 minutes. Uncover and stir until onions are almost dry (do not brown), about 5 minutes. Transfer onions to a blender. Add buttermilk and 1 tablespoon water. Purée until smooth. Season with kosher salt.
Preheat oven to 350°F. Arrange a wire rack inside a rimmed baking sheet. Melt 2 tablespoons butter with oil in a large cast-iron skillet or large heavy skillet over medium-high heat. Season steak generously with kosher salt. Cook until a deep brown crust forms, 3-4 minutes per side and 1-2 minutes on edges. Place on prepared rack. Roast until a thermometer inserted into steak registers 115°F, about 40 minutes. (Steak will carry over to medium-rare.) Let rest for 30 minutes.
Melt remaining 2 tablespoons butter in a large skillet over medium-high heat; cook until butter browns, about 2 minutes. Add steak and cook for 30 seconds per side, allowing steak to absorb butter. Cut steak into 4 slices. Place 1 steak slice on each plate and sprinkle with sea salt. Spoon warm onion purée alongside. Scatter herbs over purée.
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