What is the best way to infuse mint into a tomato sauce?

Mint likes oil. And it likes water. And it likes alcohol. Like most complex flavors, mint is complicated.

The greener, vegetal notes are going to be from compounds like chlorophyll, and will be alcohol and very weakly water soluble. The astringent, sharper notes are going to from compounds like menthol, which are oil soluble.

In general, the faster flavors, the ones that hit fast and fade faster, are water soluble, while the ones that linger are oil soluble. Mint flavor is sold both as an extract, in a base of water or alcohol, and as an essential oil, in a base of... oil.

As for cooking, I would skip the tomatoes entirely, and do a mint and parsley pesto with walnuts and olive oil. Why? I like mint.

Mint is closely related to Basil and can be treated in the same way. So for a tomato sauce you will probably get the b est results by adding bruised mint leaves towards the end of the process. Too much cooking will boil off the more subtle aromatic flavours and you will end up with something a bit harsh and medical.

It may even be best to add ripped mint leaves to the warm sauce just before serving as this will give you the subtle and fresh flavours you want without it ending up like tomato toothpaste.

I would cook the sauce and toss in the mint at the last minute, right before you use the sauce. I don't think you want to heat/cook the mint, as it might result in too much of a vegetal note. Of course, more mint and/or more surface area (chopped) would produce more flavor...depends on your use of the sauce.

Depends on what you are planning to serve the sauce with, you could consider a mint jelly which is mildly sweetened and would not overwhelm the tomato flavor , it would save you from sweetening the tomatoe sauce. If you don't have that particular item right now , go with mint leaves directly into the sauce when it starts to simmer, leave it there for a couple of minutes (2 to 5 minutes) but make sure you can remove them easily (cheese clothe, a small mint branch with leaves attached), the longer the mint simmers with the sauce, the stronger mint taste you'll get. I would not put it at the last minute as your sauce might overcook as you try to transfert mint flavor to it.

It depends on what you mean by "tomato based mint sauce." The tomato sauce that is one of the classical French mother sauces contains ingredients that probably don't pair well with mint (bay, thyme, and pork come to mind). I assume that you intend your "sauce" to be more of a simple Italian tomato sauce or perhaps a salsa?

I would definitely recommend using your mint without cooking it - I would chop it and mix it with your other ingredients immediately prior to service to take full advantage of its magnificent aroma.

If the tomato part of your dream sauce needs cooking, keep your cooking time to a minimum to make sure your tomatoes don't lose their bright acidic taste - throw-in the mint at the end. I don't think that mint would taste as good with long-cooked tomatoes (they'd be too sweet) and I really don't think your fresh mint would cook well into your tomatoes. Ultimately, I think that your fresh mint would go best with uncooked tomatoes.

For me at least, the taste of tomato and mint immediately brings to mind tabbouleh salad. Take away all the parsley and bulgur wheat (they certainly aren't very sauce-y) and the remaining ingredients really form the basis of a salsa fresca - sort of a minty pico de gallo.

Going with that idea, I'd recommend a salsa made with crushed or chopped tomatoes, minced garlic, very finely chopped onion or shallot, lemon juice, olive oil, your fresh mint, maybe some fresh basil, salt and pepper - maybe even some very fine lemon zest? Run it through a food-processor if you want it more smooth - personally, I'd leave it a little chunky so that the different ingredients surprise the palate and the aroma of the fresh mint doesn't get lost.

I think this concoction would taste great on bruschetta or crostini, on fish, maybe with poached eggs and toast?

That's the best recommendation I can make w/o knowing more about the plans you have for your tomato-mint sauce.