Heat oven to 375°F. Rub beef with 1 teaspoon oil, 1/2 teaspoon salt and 1/4 teaspoon pepper and place in a small roasting pan. Roast for 40 to 55 minutes, checking early, until the thickest center reaches 145°F.
Meanwhile, melt butter with the remaining 2 teaspoons oil in a skillet over medium-low heat. Add onions and the remaining 1/4 teaspoon salt and cook for 30 to 40 minutes, stirring often, until deeply golden and soft.
Add 2 tablespoons water to the onions and scrape up browned bits. Cook until the liquid evaporates, adding the optional 2 tablespoons water only if the pan begins to scorch before the onions soften.
Rest the roast for at least 10 minutes before slicing very thinly across the grain. Mix sour cream, horseradish, mustard and the remaining pepper.
Reduce oven to 300°F and warm split croissants cut-side up for 5 minutes. Spread horseradish cream over their cut faces.
Divide warm beef and caramelized onions among the bottoms. Close with the tops and serve promptly.