Preheat oven to 375°F (190°C). Grease a baking dish large enough to hold 4 apples with 1 tablespoon butter. Core each apple about 3/4 of the way down, leaving a bottom so filling won’t fall through. Using a melon baller or small spoon, enlarge the cavity slightly and score the outer skin lightly in a cross pattern to avoid splitting.

Make filling

In a medium bowl combine 1 cup brown sugar, 1/2 cup rolled oats, 1/4 cup flour, 1 1/2 tsp ground cinnamon and 1/4 tsp salt. Cut in 4 tablespoons cold cubed butter with a pastry cutter or fingertips until mixture resembles coarse crumbs and holds together when pressed. Stir in 1 tablespoon lemon juice and 1 teaspoon vanilla.

Stuff apples and bake

Spoon the filling tightly into each cored apple, mounding slightly. Place apples in the prepared baking dish. Add 1/4 cup water to the bottom of the dish to create steam. Bake uncovered for 30–40 minutes, until apples are tender when pierced with a knife and filling is golden on top. If tops brown too quickly, tent loosely with foil for the last 10 minutes.

Caramel Sauce

Make caramel

While apples bake, combine 3/4 cup granulated sugar and 2 tablespoons water in a medium heavy-bottomed saucepan over medium heat. Gently swirl the pan (do not stir) until sugar dissolves and begins to turn amber at the edges, about 6–8 minutes. Watch closely — once it starts browning, it will color quickly.

When sugar reaches a deep amber color, remove from heat and carefully pour in 1/2 cup warmed heavy cream (it will sputter). Whisk in 2 tablespoons butter until smooth, then stir in 1/4 tsp salt. If any sugar crystals form on the sides, return briefly to low heat and whisk until smooth. Set aside to cool slightly; sauce will thicken as it cools.

Brown-Butter Vanilla Ice Cream (quick stovetop base)

Brown the butter

In a small skillet over medium heat, melt 4 tablespoons unsalted butter and cook, swirling frequently, until solids turn golden-brown and nutty aroma develops, about 4–6 minutes. Watch carefully to avoid burning. Remove from heat and let cool slightly.

Warm dairy

In a medium saucepan combine 1 cup whole milk and 1 cup heavy cream and the cooled brown butter (including browned bits). Warm over medium heat until just steaming — do not boil.

Temper yolks

In a bowl whisk 4 egg yolks with 1/2 cup granulated sugar until pale and slightly thickened. Slowly pour about 1/3 cup of the warm milk mixture into the yolks, whisking constantly to temper. Then slowly whisk the tempered yolks back into the saucepan with the remaining milk mixture.

Cook custard

Cook the mixture over medium-low heat, stirring constantly with a wooden spoon or heatproof spatula, until it thickens slightly and coats the back of the spoon (temperature about 170–175°F / 77–80°C). Do not let it boil.

Finish and chill

Remove from heat and strain custard through a fine mesh sieve into a bowl to remove browned butter solids and any cooked bits. Stir in 1 teaspoon vanilla and 1/4 teaspoon salt. Chill the base until very cold (at least 4 hours or overnight). Once chilled, churn in an ice cream maker according to the manufacturer's instructions until soft-set, then transfer to a container and freeze until firm, about 2–4 hours. (If you do not have an ice cream maker, freeze in a shallow container, stirring vigorously every 30 minutes until firm.)

Finishing & Serving

Assemble warm layered apple

Remove baked apples from the oven. Place one warm apple on each serving plate. Spoon 1–2 tablespoons of warm caramel sauce over each apple and around the plate. Add a generous scoop (about 1/2 to 3/4 cup) of brown-butter vanilla ice cream on top or to the side of each apple so the warm fruit melts the ice cream slightly.

Garnish

Dust each serving with a light sprinkle of 1/2 teaspoon ground cinnamon (divided between plates) and scatter toasted chopped pecans or walnuts if using. Serve immediately so the contrast between warm apple, hot caramel and cold ice cream is preserved.