Preheat oven to 400°F. Unroll puff pastry and place on a parchment-lined baking sheet. Score a ½-inch border around the edge (don’t cut all the way through).
Heat jam in a pan over a low heat for 30 seconds, just to loosen, stirring occasionally. Finely slice the apples into rounds, discarding any seeds (use a mandoline, with a guard, if you have one).
Toss the apple with the apricot jam and arrange nicely, in layers, inside the border, hiding any smaller tattier bits of apple underneath. Cook on bottom rack of oven for 30 minutes, or until beautifully golden and cooked through, scattering over almonds for the last 5 minutes.
Slice up and serve topped with balls of vanilla ice cream.