At least 1 day before serving, proceed through Step 3. Line a colander and a medium-size strainer with a sturdy paper towel; set each in a bowl.
Pulse strawberries and granulated sugar in food processor until coarsely chopped. Scrape into colander, top with a paper towel and refrigerate overnight to drain well (this is important). Clean processor.
Put ricotta, cream cheese and confectioners' sugar in processor; pulse until smooth. Transfer to strainer, cover with a paper towel and refrigerate overnight.
Just before serving: Prop cones upright in tall glasses partially filled with granulated sugar to support cones. Fold drained berries and almond extract into ricotta mixture. Spoon into a gallon-size ziptop bag. Cut 1⁄2 inches off a corner; pipe filling into cones.