Preheat the oven to 375 degrees F. Put the beet on a sheet of foil, drizzle with the olive oil and wrap to enclose. Bake until easily pierced with a knife, 60 to 70 minutes. Let cool, then peel and chop.
Bake the phyllo shells as the label directs. Let cool.
Pulse the goat cheese and heavy cream in a mini food processor until smooth. Add the chopped beet, honey and 1/4 teaspoon salt. Pulse until very smooth and bright pink, about 1 minute.
Transfer the beet mixture to a large resealable plastic bag and snip off a corner; pipe into the phyllo shells. Refrigerate at least 20 minutes before serving.