In a large saute pan over medium heat, melt 2 tablespoons butter. Add apples and saute, turning occasionally for even cooking, until slightly golden and softened but still holding its shape, about 10 to 12 minutes.
Meanwhile, in a small saucepan over medium heat, add sugar and water. Stir occasionally and watch closely until color just starts to change to golden-amber, about 6 to 8 minutes. Remove pan from heat and carefully add cream (mixture will bubble and release steam). Gently stir until bubbles subside slightly, and put back over very low heat. Stir in remaining tablespoon butter, vanilla, and salt, and remove from heat. Set aside to thicken slightly.
Disperse apples into 4 serving bowls and spoon warm caramel sauce over to cover. Top each with a scoop of ice cream, spoon more caramel over the top, and garnish with peanuts.