Make the compote: Combine the tarragon sprigs, granulated sugar and 1 3/4 cups water in a medium saucepan; bring to a boil and stir until the sugar dissolves. Add the rhubarb, reduce the heat to low and cook until softened but not falling apart, about 3 minutes. Strain the rhubarb over a bowl, reserving the liquid (keep the rhubarb in the strainer to drain completely while assembling the strudel).
Make the strudels: Preheat the oven to 425 degrees F. Pulse the cream cheese, confectioners' sugar, flour and vanilla seeds in a food processor until just combined.
Lay out 1 phyllo sheet on a clean dry surface (keep the remaining phyllo covered with a slightly damp kitchen towel). Brush lightly with clarified butter and sprinkle with 1 teaspoon chopped tarragon and 1/2 teaspoon granulated sugar. Layer another sheet of phyllo on top and top with more butter, tarragon and granulated sugar. Repeat the layering with the remaining phyllo; do not top the final sheet. Cut the stack of phyllo in half crosswise. Reserve the remaining clarified butter (about 1 tablespoon).
Put each phyllo stack on a piece of parchment paper that's slightly larger than the stack. Position the stacks with the long sides facing you. Divide the cream cheese mixture between the stacks, spreading it in a 2-inch-wide strip along the edge closest to you; leave a 1-inch border at the bottom and sides. Spoon the rhubarb pieces over the cream cheese filling. Starting from the side closest to you, use the parchment to lift the phyllo and roll tightly around the filling to make 2 logs; arrange seam-side down. Transfer the strudels (on the parchment) to a baking sheet.
Brush the strudels with the reserved clarified butter and sprinkle generously with granulated sugar. Bake until golden brown and crisp, 15 to 20 minutes. Meanwhile, bring the reserved rhubarb liquid to a boil in a saucepan. Reduce the heat and simmer, stirring occasionally, until syrupy, about 15 minutes.
Let the strudels cool 10 minutes on the baking sheet. Slide the strudels (on the parchment) onto a rack; let cool 45 minutes. Sprinkle with confectioners' sugar and drizzle with the rhubarb syrup.
How to clarify butter:
Strudel recipes often call for clarified butter, which helps prevent the dough from getting soggy. To make it, melt the butter over medium-low heat until the milk solids separate and rise to the top, 3 to 5 minutes. Let cool, then skim off the white foam and discard. Spoon the remaining clarified (clear) butter into a bowl; discard the solids left at the bottom of the pan.