Mix the flaxseed meal, bouillon, and water. Let it sit for 5 minutes to thicken.
Meanwhile, add the chickpeas, oat flour, poultry seasoning, and garlic powder to a food processor. Pulse this together until it's a thick paste. Add the flax mixture to the processor and pulse until it's combined.
Transfer this chickpea mixture to a bowl. Cover it and chill it for 15 minutes. This will let it set and give time for the oat flour to absorb the extra liquid.
Divide the chickpea mixture into thirds, about 1/2 cup each, on parchment paper. Roll the portions into rounded cylinders and then flatten them to a thickness of about 1/2 an inch. Flip them over to make sure they aren't stuck to the parchment paper.
Trim the vegan ham and cheese slices, if needed, and place them on the chickpea slabs in two layers. If the vegan cheese slices are rigid, as they usually are, slice them into strips. This will help when rolling them. Roll them into logs and press to seal the edges and ends.
Add a thin layer of oil to a large pan and heat it to medium. Prepare one shallow dish with melted vegan butter and another shallow dish with breadcrumbs. Roll the cylinders in the melted butter, then coat them with the breadcrumbs.
Add them to the sizzling hot oil and let them cook until they are golden brown on all sides. This should only take a few minutes in total. Let them cool down before slicing into them. Garnish them with parsley and enjoy them as is or with your favorite sauce or gravy!