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What you've got is a set of flavors and a desire for a certain texture. That texture should be supplied by something which provides protein as well as body.

The classic non-meat patties are made of:

lentils
chickpeas (garbanzos, ceci)
beans (including soybeans (edamame)
nuts
root vegetables (including potatoes, turnips, yams, tapioca)
stalk vegetables (brassicas, mostly)
and grains

None of these contain gluten except grains; the grains that have gluten for the purposes of celiac disease are wheat, barley and rye. If you have some other kind of gluten intolerance you should specify that.

There are an infinite number of recipes, and any vegetarian cookbook should have several suggestions.