In a medium saucepan, combine the milk, brown sugar, cocoa powder, cornstarch, and salt. Whisk constantly over medium heat until thickened and no lumps remain, about 5 minutes. Reduce the heat as necessary to avoid burning on the bottom of the pan.
Remove the pan from the heat and stir in the yogurt and vanilla.
Divide the mixture between popsicle molds and insert the sticks. Freeze until firm, at least 4 hours and up to overnight.
Notes
TO STORE: Homemade fudgsicles will keep in the freezer for up to a month. It's best to store them in the molds; if you have to remove them, wrap the pops individually in plastic wrap, then place them in an airtight container or freezer bag.