Combine the almond paste and sugar in a medium bowl and beat with an electric mixer until the mixture is completely smooth. Add the eggs one at a time, beating well after each addition. Beat in the almond extract and salt.
Bring the half and half to a simmer in a heavy medium saucepan. Slowly beat the hot half and half into the almond paste mixture. Pour the entire mixture back into the pan and place over low heat. Stir constantly with a whisk or wooden spoon until the custard thickens slightly. Be careful not to let the mixture boil or the eggs will scramble.
Remove from the heat and pour the hot almond custard through a stainer into a large, clean bowl. Allow the custard to cool slightly, then stir in the heavy cream. Cover and refrigerate until cold or overnight.
Stir the chilled custard, then freeze in 1 or 2 batches in your ice cream machine according to the manufacturer's instructions. When finished, the ice cream will be soft but ready to eat. For firmer ice cream, transfer to a freezer-safe container and freeze at least 2 hours.