In a large bowl, whisk whole-wheat flour, all-purpose flour, and salt. Cut in cold butter with a pastry cutter or fingertips until pea-sized bits form. Beat the egg with 3 tablespoons ice water, drizzle in, and mix just until the dough holds together. Add up to 2 more tablespoons water if dry. Press into a disc, wrap, and chill 30 minutes.
Lightly grease a standard 12-cup muffin tin. If using a nonstick dark pan, reduce bake time by a couple minutes to avoid over-browning.
In a saucepan over medium heat, melt butter with olive oil. Add onion and a pinch of salt; cook 3–4 minutes until translucent. Stir in garlic for 30 seconds.
Sprinkle flour over onions, stir for 1 minute. Whisk in chicken broth gradually, then milk and cream. Simmer 2–3 minutes until thick enough to coat a spoon.
Stir in chicken, mixed veggies, potatoes (if using), thyme, parsley, salt, pepper, and paprika. Taste and adjust seasoning. Let cool 10 minutes so it thickens—no soggy bottoms, please.
On a lightly floured surface, roll to about 1/8 inch thick. Cut 12 rounds about 3.5–4 inches wide for bottoms and 12 smaller rounds or decorative shapes for tops. Re-roll scraps as needed.
Press larger rounds into muffin cups, letting dough come just above the rim. Spoon filling into each, nearly to the top but not overflowing. Cap with smaller rounds or lattice strips; crimp edges with a fork. Cut a small vent if using full caps. Brush with egg wash and sprinkle flaky salt if you're feeling fancy.
Preheat oven to 400°F (200°C). Bake 18–22 minutes until tops are golden and the filling bubbles at the edges.
Cool 10 minutes. Run a thin knife around edges and lift out. Resist the urge to inhale immediately—molten pie lava is not your friend. Serve alongside our Homemade Hummus with carrot sticks for an epic starter-plus-main combo.