Lots of things can be made as finger food which can be picked up whole but turn to mush quite easily in the mouth. For instance sweet potato wedges/batons roasted in the oven, the outside is 'dry' for picking up but the inside is soft like puree/mash. You can also steam them although they can be a bit slippery to pick up as finger food, if you find this try cutting in a crescent moon shape or in large rounds with a hole in the middle, it gives a grasping point for small hands. You can even just squidge a bit of a notch in once it's cooked.
Try oven baked or steamed vegetables such as carrots, parsnip, turnip, butternut squash
Fruits such as apple and pear
Fresh fruits such as melon, nectarine, mango, kiwi tend to mush easily if they are ripe.
As for eggs, my DS doesn't like the texture of scrambled egg but will eat omelet happily. Maybe worth a try, it's still soft. or hard boiled, both parts are soft but the yolk especially is easily mushed down with gums.