Burgundy is in a funny place right now. It used to be that only those really passionate wanted it, and so Paris was a Bordeaux-based city in the restaurants except for the luxury restaurants, which carried the most prestigious bottles. Now there is unprecedented worldwide demand for Burgundy and there have been some very small (quantity-wise) harvests, and so prices are going through the roof. But at least you’ll find Burgundy on almost all restaurant lists, and at prices that are amazingly good compared to, say London or NYC, FWIW.
In addition to Les Climats, I would suggest favorites of Parn and mine, Aux Crus de Bourgogne, Auberge Bressanne, Aux Bons Crus, and Les Marches. They’re all under the same ownership and have similar concepts and quite a bit of overlap in their excellently-chosen wine lists. Aux Crus de Bourgogne will give you the widest choice of Burgundy followed by Auberge de Bressanne, then the other two. I’ve never been to Les 110 de Taillevent, but I imagine you can get a good selection there – at a price.
Given the current state of Burgundy, a little advice: global warming is having a strong effect in Burgundy as it is in vineyards everywhere. As a result, vineyards that traditionally are less-favored because of their cold climates are now coming into their own. This is particularly true in the very hot 2018-2019-2020 vintages, which are mostly what are on restaurant wine lists these days. It’s worth your while to find Bourgogne, and particularly Bourgogne Hautes Côtes de Nuits and Hautes Côtes de Beaune and wines from the Côte Chalonnaise from good producers – both colors. I’ve been enjoying a lot of them. Also, some amazingly fine wines are coming from the Mâconnais these days.
If you want to see what classic red and white Burgundy is like, 2017 is what you have to seek out. The wines have not shut down (Burgundy often “goes to sleep” after three years and may not wake up until 15, even 20, years after the vintage) and are drinking beautifully right now with all the freshness and precision that one often loses in 2018-19-20 at the expense of the voluptuous fruit. Alas, though, 2017s are rapidly disappearing from the wine lists. But above all, remember the most important rule of Burgundy: producer, producer, producer. A modest wine from a good producer (and just because you haven’t heard of a producer doesn’t mean that s/he isn’t good) is almost assuredly better than a grand appellation from a mediocre producer.
Also, before you accept a wine, check the stated alcohol – I do not like 14% and would refuse any Burgundy with a stated alcohol in excess of 14% (and there are many in 2018-19-20).
Finally, make sure that you drink the wines at cellar temperature 13-15ºC, 55-60ºF – often the reds, especially, are served too warm. If you get a wine that is too warm, don’t hesitate to ask for a bucket (“un sceau” in French, pronounced pretty much like the English word “so”) to cool the bottle off (doesn’t do you much good if it’s a wine by the glass).