There is no lack of shopping opportunities in Burma. But what do you need to bring back?

Cosmetics

Thanaka

In terms of cosmetics, clearly Thanaka is at the top of the list. It is a yellow paste used as sunscreen by the Burmese. It exists in different versions, but the compact powder version is the easiest for foreigners to use. The “wood” version is the cheapest but requires a stone base.
When you are in Burma, imitate the Burmese by putting Thanaka on both your cheeks to protect you from the sun, it will protect you from sunburn and will also attract the sympathy of the locals. Do not put too much, a thin layer will be enough. You can put some on the whole face but it will not be very aesthetic. You can also draw patterns using a toothpick or your nails. When you return to France, use the Thanaka as a moisturizing, purifying mask.

Where to buy it? In all the stores around the pagodas, temples. A jar, with “export” marked on it, costs in the 1000kyats.

Thai and Korean cosmetics

Go to UNIQ HAN, Karmart for a selection of Thai products designed in Korea (such as Cathy Doll, Korean) and a multitude of aloe vera gels.
For real Korean products made in Korea, you have the stores of Holika Holika, Etude House and Tony Moly at the Diamond Plaza shopping center. The prices are more expensive than in Korea for sure, between +20% and twice as expensive (so you might as well buy in France). Nature Republic has a store on 73rd Street (crossing with 36th Street) in Mandalay, but prices vary according to the product. In general, the masks are only 20% more expensive than in Korea, which justifies making a stock of them.
Where to buy: Diamond Plaza, UNIQ HAN, Karmart

Souvenirs

Many souvenirs are made with teak wood: Buddha masks, elephant statues, Buddhist steles…
It is just necessary to be careful with the masks of Buddha if you pass by Thailand which does not appreciate that Buddha is used as object of decoration.
You may be interested in wooden postcards, or puppets.
Burmese cigars can be found at local markets (cheaper) or in cigar factories (on the inle Lake). Cigars are purchased in sets of 25 and not by the unit. If you forgot to buy some, you can always find some at the airport.
Paper umbrellas are another marvel of Burmese craftsmanship. Depending on the size of the umbrella, you can pay between 1$ and 20$
Bowls made of lacquer or horse hair are particularly popular in Bagan. You can visit workshops near New Bagan. The price varies greatly depending on the size of the bowl but be aware that lacquer is an expensive ingredient.

Clothing

Longyi are Burmese skirts made with a single piece of fabric. They are made for both women and men, the only difference is the way of tying it (and the fabric too, men don’t wear shiny fabrics).
Where to buy it? Go to the markets where they will make custom longyi for 5000 kyats (in cotton) or more expensive if it is in silk. It’s up to you to choose the fabric that suits you best. Avoid buying them in stores near the pagodas.
For non-traditional but cute clothes, go to the Diamond Plaza, you will find super cute dresses or skirts between 6€ and 10€. Another even cheaper address is 73rd Street near Mandalay University.
Wherever you go, haggle!