I'm also lacto-ovo vegetarian, and then my twins ended up being intolerant to dairy. I was able to exclusively nurse them both while being basically vegan + eggs. So it's very possible.
It was similar to being pregnant, I focused on trying to make each thing I ate really count. I used Earth Balance instead of butter, cooked with oils, used soy or cashew milk in cooking, and learned a bigger repertoire of vegan meals. I put all the easy vegan meals that we tried onto index cards and kept them in a little booklet in the kitchen, which really helped meal planning be quick and easy. I also started buying nuts like almonds and cashews in bulk so I could snack on them. Lots of protein and healthy fats in those! My main focus was on making sure I was getting lots of protein, iron, and fats. Of course, I was nursing 2 infants, so you'll have a little more leeway.

I actually had more trouble with the sweets than with the meals. If you like chocolate, you'll be happy to know that dark chocolate has no dairy. Also, I thought the chocolate rice ice cream was better than any of the soy ice creams, in case you get a craving for that.
Cheeses: I found that it wasn't too hard to cut cheese out. You can easily make a tofu "ricotta" that works perfectly in lasagna, on pizza, and in something like a baked ziti casserole. Tons of recipes for that out there if you google, here are some: the
Veganomicon one we use, and
another I found. I think the nutritional yeast changes the flavor too much, so I usually skip that. The Veganomicon actually has a whole bunch of sauces and things that you can substitute for cheese. I have a vegetarian crockpot cookbook that has a vegan mac & cheese recipe that is really good.
You can try different pre-packaged soy cheeses, although I didn't find any I really liked. Just be careful that they are
dairy-free because some of the soy cheese has dairy (without lactose). Be on the lookout for
casein or whey, which are dairy ingredients.
I'd suggest that you start by going through your staple and favorite recipes and seeing if there is anything that you can do to change those to vegan. Then look through your favorite cookbooks for vegan recipes that you maybe haven't tried yet.
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