Heat the oven to 400°F (200°C). Line 2 baking sheets with parchment paper.
Combine the milk, water, oil, and salt in a saucepan and bring just to a boil over medium heat. Remove from the heat as soon as it bubbles.
In a large bowl, whisk together the sour tapioca starch and sweet tapioca starch. Pour the hot liquid over the starches and stir with a sturdy spoon until evenly moistened and shaggy. Let cool until warm to the touch, about 10 minutes.
Add the lightly beaten eggs in 2–3 additions, mixing well after each until fully absorbed; the dough will be sticky and elastic.
Add the finely grated Minas cheese and knead in the bowl until the dough is smooth, glossy, and pulls away from the sides, 2–3 minutes. If too soft to handle, chill for 10 minutes.
Scoop heaping tablespoon portions (about 1–1.25 inches wide) and roll into balls with lightly oiled hands. Arrange on the prepared sheets, spacing about 1.5 inches apart.
Bake until puffed, set, and lightly golden in spots, 18–22 minutes. The bottoms should sound hollow when tapped. Serve hot.