Melt the oil in a wide skillet over a medium flame until it shimmers. Add the cheese cubes in a single layer, turn the heat down to medium-low, and let the cheese brown on the first side, 3-5 minutes. It will spit and hiss, so be careful. Loosen the cheese with a thin, metal spatula, and flip each cube onto a second side. Let brown on the second side, 3-5 more minutes, then sprinkle with a bit of salt and remove the cheese to drain on a double layer of paper towels.
If the oil in the pan burned or smoked, pour it out, wipe out the pan, and add another tablespoon of oil; otherwise, you can leave the oil in the pan for the next step.
Use a sharp paring knife to slice the chard leaves off of the stems. Trim away the tips of the stems and rinse them well, rubbing off any sand. Slice each stem in half, then chop into 1/4" lengths. Set the stems aside. Give the chard leaves a rough chop, then soak them in a large bowl of cool water, swishing occasionally to loosen any dirt or sand clinging to them.
Heat the oil over a medium flame until it shimmers, then add the chard stems, onion, ginger, cumin, garlic, chile flakes and turmeric. Saute, stirring occasionally, 5 minutes, then add 1/4 cup water and sprinkle with 1/4 teaspoon salt. Cover the pan and let the mixture sweat until the chard stems are tender, about 10 minutes, stirring occasionally.
Meanwhile, lift the chard leaves out of their soaking water and place them in a large saucepan or soup pot with water still clinging to them. Cover the pot and place it over a medium flame. Steam the chard until the leaves are just wilted and bright green; this will only take a few minutes. Scoop the leaves into a strainer, rinse with cool water, and squeeze out most of their liquid. (Alternately, chop the raw leaves and add them directly to the stems, cover with a lid, and cook, stirring occasionally, until the leaves are tender, 10-15 minutes.)
Chop the leaves fairly finely, then stir them into the onion mixture. Stir in the yogurt, milk, and cheese cubes, gently heat through over a low flame, then remove from the heat and add the lemon juice. Taste, adding more salt or lemon juice if needed to bring up the flavors.
Serve the saag paneer over rice, garnished with cilantro and lemon wedges to squeeze over the top