Prepare popcorn using oil, as directed in your popper, or place oil in a large pot with a lid. Add kernels and cover. Place over medium to medium-high heat. Shake occasionally to prevent burning. When popping slows to 3-5 seconds between pops, remove pot from burner. Allow to cool 1-2 minutes. Meanwhile, bring cream and milk to a simmer. Add the popcorn and simmer 10 minutes. Pass through a fine-mesh sieve or chinois. Add sugar, salt and extract and stir to dissolve. Cover tightly and refrigerate 3-4 hours.
Pour base into ice cream freezer; freeze according to manufacturer instructions. Transfer to a container which has been chilled in the freezer. Place a layer of plastic wrap over the ice cream before covering tightly. Freeze until hard, 2-4 hours, before serving.
For the sauce, place water and sugar in a medium-large saucepan over medium heat. When sugar is dissolved, turn the heat up to medium-high and simmer about 10 minutes, until it turns amber. Watch closely, as it can burn quickly. When sufficiently brown, turn heat down to medium-low and add cream; it will bubble up, so use a large enough pot to prevent spillovers. Off heat, stir in butter and salt. Serve over buttered popcorn ice cream.