In the medieval period which would have been the more profitable activity- rearing cattle for meat, or rearing cattle for milk ? And which would have required more investment?

This is just an educated guess, but probably milk. 

The problem with farming for meat is that prior to the agricultural revolution of the 18th century, something I talk about in my economic development series in reference to growing cereal crops, it was difficult to keep meat animals alive over the winter due to lack of food for them. Hence, you’d slaughter animals when they were roughly a year old, which limited their growth and thus how much meat you could get from them. (Lack of refrigeration and other means of storage was also a problem.)

During the agricultural revolution of the 18th century, you have a number of interlocking developments that changed that:

1. Changing rotation of crops to replace fallow fields with turnips, clover, hay, and other legumes. Not only do these help restore nutrients to the soil, which improves the yield of cereal crops, but they also meant that you now had a new source of fodder to feed your animals and could afford to keep them alive over the winter.
2. Introduction of water-meadows. Water-meadows are areas of grassland that are irrigated to keep them continuously damp, which promotes the early growth of grass, which allowed for animals to be pastured on these meadows in those tricky periods of late winter/early spring where you’ve run out of winter fodder but the legumes and grass haven’t kicked in generally. 
3. Selective breeding of livestock. By carefully breeding animals for specific traits, farmers profoundly reshaped the economic potential of entire species. To give an example, the “average weight of a bull sold for slaughter at Smithfield was reported around 1700 as 370 pounds (170 kg), though this is considered a low estimate: by 1786, weights of 840 pounds (380 kg) were reported.” 

The combination of these factors meant that animals were now routinely being kept alive over the winter, which means they produced much, much more meat, especially once farmers created breeds of cattle especially for that purpose. 

And when I say more meat, I mean cows that looked like this:

That’s a square cow. That thing couldn’t be more genetically modified unless it came shaped in detachable burger shapes.