Use a wine you would drink. A dry, full-bodied red, a red Burgundy is classic, and it does not need to be expensive but it does need to be one you like, because the long reduction concentrates everything. Avoid bottles labelled "cooking wine", which are salted and flat.
This is a pure-wine braise. There is no stock, which gives a deeper, more assertive sauce. If you prefer it gentler, hold back a splash of the wine and make it up with a little chicken stock.
Choose one braise method, not both. Finish it on the hob or in the oven, whichever suits, for the same one to one and a half hours. The two steps are alternatives, not a sequence.
Skim the fat and thicken if needed. Spoon off any fat that gathers on top before serving. If the sauce is thin, simmer it uncovered to reduce, or stir in a little butter-and-flour paste.