Here's a website that goes into it and includes this picture of the gizmo to which you refer:
This little single serving device is meant to fit over a cup, hold about two tablespoons worth of coffee, and the required amount of water for brewing. There is a screen on the bottom, and a plunger with another screen on it that screws down on top of the coffee to smush it between the two filters. The tightness of this top filter controls the strength of the coffee. The tighter it is, the slower water flows, and the stronger the brew.
EDIT: Regarding the "bonus question": I'm not aware of chicory as an ingredient in Vietnamese/Thai iced coffee, I'm only familiar with it as a traditional New Orleans addition, having migrated from France. It became a popular substitution during the French Revolution when coffee became prohibitively expensive. Certainly the website I linked to uses it, so it may be traditional for some. Certainly the French significantly influenced Vietnamese and Thai cuisine. I dug up a recipe for New Orleans chicory coffee:
1 lb. coarsely ground coffee,
1 1/2 oz. roasted and chopped chicory,
2 1/2 quarts water,
3 oz. simple syrup,
Large Mason jar,
Combine the ground coffee, chicory and water in a stockpot. Stir with a wooden spoon, cover and let steep at room temperature for 8-12 hours.
Carefully break the crust of the coffee grounds with a spoon and strain through a fine-mesh sieve into the Mason jar. Add simple syrup to concentrate and stir to combine.
Serve over ice and add milk to taste—most people opt for about a 50/50 ratio of milk to chicory-coffee concentrate.
Keep refrigerated and use within 1-2 days. Yields 4-5 cups of concentrate.