In the bowl of a stand mixer, cream together the butter and sugar on medium-low speed with the paddle attachment. Add the eggs one at a time and mix until combined. Add in the vanilla extract, baking powder, cinnamon, and salt, scrape down the sides as needed until combined.
Add in the flour a little at a time until fully incorporated, but do not overmix. Scrape down the sides as needed. Cover with plastic wrap and place in the fridge for one hour or up to overnight.
Preheat the oven to 350°F. Roll the dough into 2-inch sized balls, about 2 tablespoons each. Place on an ungreased cookie sheet about 2 inches apart. Bake for 12-15 minutes until slightly golden brown around the edges. Let cool slightly on the cookie sheet for about 3-5 minutes, transfer tea cakes to a wired rack to cool completely.
Cinnamon glaze
While the tea cakes are cooling, make the glaze by whisking together the powdered sugar, vanilla extract, and ground cinnamon in a medium-sized bowl.
Add in 3 tablespoons of the milk and whisk to combine. If the glaze is too thick (not pourable) add another 1 tablespoon of milk. Evenly glaze the tea cakes while still on the wired rack, over a cookie sheet to catch any excess glaze. Let the glaze set up for a few minutes and enjoy.