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Preheat the oven to 375°F. Lightly grease a couple of baking sheets; or line with parchment.
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Put the butter, water, milk, and salt in a saucepan, and heat until the butter has melted and the mixture has come to a full boil. While the mixture is heating, put 2 cups of tapioca flour into a mixing bowl.
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Pour the boiling butter mixture over the tapioca flour, beating to combine. Beat at high speed until the mixture becomes smooth and elastic-looking; this will happen very quickly.
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Beat the garlic and cheese into the dough until well combined.
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Stick your finger into the dough. If it's uncomfortably hot, let it sit for a couple of minutes to cool a bit; you don't want to cook the eggs when you beat them in. If it seems hot but not burning hot, continue with the next step.
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With the mixer going, gradually dribble in the beaten eggs, beating until well combined and smooth. We've found that tapioca starch/flour varies in its absorption capabilities. If your batter is thick but pourable, like cake or pancake batter, beat in additional tapioca flour/starch until it's thick enough to hold its shape when you scoop some onto a baking sheet. It should be the approximate consistency of cream puff batter: when you plop it onto the pan, it should settle slightly, but not spread into a puddle.
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Drop the mixture in 2-tablespoon balls (about the size of a golf ball) onto the prepared baking sheets, spacing them about 1 1/2" apart. A level tablespoon cookie scoop works well here. The buns will settle slightly as they sit.
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Bake the buns for about 20 minutes, until they have a freckled appearance (from the browning cheese), and they're beginning to color a bit. Remove them from the oven, and serve hot.
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Wrap any leftovers in plastic. Store at room temperature for a couple of days; freeze for longer storage. Rewarm briefly (the microwave is fine) before serving.
