Heat oven to 375 degrees. Set twelve 4-inch bottomless tartlet (flan) rings or twelve 4-inch tartlet pans with removable bottoms on two parchment-lined baking sheets. Lightly flour a clean working surface. Roll pate brisee to an 1/8-inch thickness. Cut out twelve 5-inch diameter circles, and fit into tartlet rings. Using a fork, prick pastry all over; chill until firm, at least 15 minutes.
Line shells with aluminum foil; fill with dry beans or pie weights. Transfer baking sheets to oven; bake shells 12 minutes, rotating once during baking. Remove beans and foil. Continue baking until shells are slightly golden, 7 to 9 minutes. Remove baking sheets from oven; transfer to a wire rack. Reduce the oven temperature to 325 degrees.
In a medium skillet over medium-low heat, melt butter. Add shallots and sugar, and season with salt and pepper. Cook, stirring frequently, until shallots are very tender and have deeply caramelized, 30 to 35 minutes. Remove from heat; set aside.
Place goat cheese in bowl of a food processor; puree until very smooth. Add heavy cream and sour cream; process until well combined. Transfer to a medium bowl.
Stir in thyme and parsley; season with pepper. Spoon filling into each shell to within 1/8 inch from top, about 2 to 3 tablespoons per shell. Arrange 1 heaping teaspoon reserved shallots on top of each tartlet.
Transfer baking sheets to oven; bake tartlets until filling has set, 13 to 15 minutes, rotating once. Transfer sheets to a wire rack to cool slightly, about 10 minutes. Remove rings. Sprinkle remaining parsley over tartlets. To serve, place tartlets on plates. Garnish each plate with frisee.