Preheat the oven to 400˚F (200˚C).
In an electric mixer with a paddle attachment or with an electric hand mixer, beat the almond paste with 3 tablespoons (39 grams) of the sugar until combined.
Add 4 tablespoons (56 grams) of the butter and mix well.
Add 1 egg, then the almond extract. Sprinkle in the flour, mix again, and set aside.
Cut out four 6-inch (15 cm) circles of puff pastry.
Trace a 5-inch (13 cm) diameter circle inside each tart, cutting slightly into the pastry but not all the way through. Prick the interior of the circle with a fork to prevent it from rising.
Divide the almond paste mixture evenly among the 4 tarts and spread on the inner 5-inch (13 cm) circles.
Peel and core the apples. Slice the apples into thin circles, preferably with a mandoline, and arrange them overlapping all but 1/2 inch (1 cm) on the frangipane. (Each tart should get at least 1 whole apple.)
Melt the remaining 4 tablespoons (56 grams) of butter. Brush the apples with melted butter and generously sprinkle with the remaining sugar.
Whisk together the egg yolk and the cream. Brush the outside rim of the tart with the mixture.
Bake at 400˚F (200˚C) for 20 minutes, then reduce the temperature to 325˚F (170˚C) for 20 to 25 more minutes or until the tarts are golden brown and the puff pastry is cooked all the way through.
Cut the tarts into 6 pie wedges each and serve 3 per person.