While many bread and pastry products do depend critically on the formation and management of gluten from wheat flours, this is not universally true.
Some types of pastry have structure dependent more on the starch networks which is the other major component of wheat flours; the texture and properties of these pastries is often dependent on the gross mechanical manipulation of the structures, depending on how the product is manipulated.
American style pie crust is a case in point. Flaky pie crust is generally manipulated in a manner which minimizes gluten production: low hydration, resting periods, and minimal mechanical manipulation.
The structure and flakiness emerges because the dough is essentially a series of butter flakes or bits, coated in starch. When baked, the starches gelate and steam from the dough pushes the butter pockets apart before the butter is fully melted and integrates into the dough structure. This is not dependent on gluten formation.
Taking La Cucina Italiana's pasta frola recipe as typical, it appears that pasta frola has more in common with flaky pie crust than it does with bread or puff pastry, where gluten formation is key.
The egg in the dough will contribute mostly water (eggs are 75% water). The remainder is primarily proteins (from the albumen in the egg white, and fats from the yolk).
This will somewhat tenderize the final pastry, and contribute an eggy flavor. It certainly does interfere with gluten formation, helping prevent a tough outcome, except in this application, gluten formation simply is not key. It will also contribute to the final color, both from the yellow pigments in the yolk, and by slightly promoting browning.
I'm going to assume that the primary question is "How can I make a flaky pie crust without wheat flour?" Note that recipe requests are off-topic on this site, but substitution questions are on-topic.
The key thing which wheat flour gives you is gluten. Quinoa is gluten-free, so it's not going to help you much. However, there are ways of making gluten-free breads, cakes, etc. The first hit which Google gives me uses cornstarch, xanthan gum, and gluten-free flours. The second uses arrowroot starch, xanthan, and gluten-free flours. Xanthan isn't the only way of substituting for gluten, but it seems promising.
Best Answer
There's a recipe I saw a while ago that uses butter that's thoroughly incorporated into the flour instead of the usual cold chunks. I think it creates a butter/flour paste, then cuts fresh flour into it just prior to adding the water (the idea being that instead of layers of tricky-to-work-with pure butter separating layers of water-dough, you would have more workable layers of butter-flour paste separating layers of water-dough).
In that case, assuming your mention of 'the dough coming together before you added water', means you have mostly butter and flour in the bowl at this point, you might add more butter to it so the ratios work out, and treat your whole first batch like that butter/flour paste. You will need to cut fresh flour into it, since the paste is only supposed to contain 2/3 of the recipe's flour and all the butter, so it might take some math to figure out how to make what you already have fit into the ratios neatly. Then add the water and gently bring it together into a dough, refrigerate for a couple hours, and roll as normal.
This might still work even if you've added water, maybe making a smaller portion of the butter/flour paste to cut into the dough, and adding a balancing amount of flour and water afterwards - you should still get some of the texture benefits, although not as much as if the whole batch were done the same way.
You will end up with more dough in either case, but you can either chill the leftover dough, or pre-divide and store the leftover portion of the butter-flour mix, and it should store pretty well (both flour and butter store well in the fridge).
(I have found the original recipe, it can be found here, and the sciency-rationale over here, at seriouseats.com)