After you've washed and chopped the strawberries, put them in a pan with the sugar and balsamic vinegar, on medium heat.

Stir everything together so strawberries are evenly coated.

The sugar should completely absorb and some liquid will cook out of the strawberries. Side note---I was worried about liquid, since the recipe didn't call for water; nothing to be concerned about, the liquid will just appear. Continue cooking for about 20 minutes, stirring occasionally.

A good portion of the time, the mixture will be lightly boiling. Make sure it doesn't burn and stick to the bottom. Add a splash of lemon juice towards the end, if you feel like it.

After about 20 minutes (more like 25 in my case), it should have reduced to the consistency of thick honey. Remove it from the burner and let it cool completely. Not sure mine got quite thick enough but it was starting to burn on the bottom and I didn't want that.  The whole thing took maybe 30-45 minutes from start to finish, and I wasn't in a rush at all.

Put in jars, then flip the jars upside down and let them cool.  This is a European canning method, using the heat of the jam to seal the jars.  I wouldn't trust it as much as true hot water or pressure cooker canning, so I store mine in the fridge and try to use it within about 6 months.  No idea if that's necessary, but just keeps my mind at ease (particularly if you're giving them away).