Put the ricotta in a fine-mesh sieve set over a bowl. Refrigerate 30 minutes to drain.
Transfer the ricotta to a large bowl. Add the cream cheese, confectioners' sugar, vanilla and almond extracts, and the orange zest and beat with a mixer until smooth and fluffy, about 1 minute. Fold in 3 tablespoons chopped chocolate. Cover and refrigerate until thick and cold, at least 1 hour.
Transfer the ricotta mixture to a resealable plastic bag. Snip off one corner and pipe the mixture into the cones. Gently press the remaining chocolate and the pistachios into the ricotta mixture. Dust the cones with confectioners' sugar.