Peel your mangoes, then slice the fruit around the pit. Dice the mango into small cubes; they do not have to be even or perfect. Add the diced mango to a large nonreactive bowl. Add the salt to the mango and toss until everything is well-coated. Cover the bowl and refrigerate for 1 day.
After the mango has cured in the fridge, over medium-low heat add oil to a large pot or deep sauté pan. Add the mustard seeds to the oil, and when they begin to make popping sounds, add the finely minced garlic and diced chili. Sauté until softened and fragrant, but before anything begins to brown, about 2-3 minutes. Add the remaining spices: turmeric, fenugreek, coriander, cumin and cayenne. Stir and sauté for an additional minute.
Add the mango, brown sugar and water to the pot. Stir, increase the heat and bring the liquid up to a simmer. Simmer for 5-6 minutes, or until the mango has softened and the liquid has slightly reduced. Turn off the heat and add the vinegar to the mango mixture. Taste and adjust to your liking by adding more vinegar, sugar, salt or spices if needed.
Using an immersion blender or blender, puree the mango to the desired consistency. I like mine a little chunky with about half of the mango pieces still intact. If you would like your amba smoother, purée it for longer and add water to thin it out. Note that amba will thicken slightly as it cools.
Once cooled, transfer the amba to jars and refrigerate. Amba keeps well in the fridge for about 2-3 weeks. Makes 3 pints.