Combine warm water, yeast, and sugar in a bowl; let stand for 5 minutes to activate the yeast.

Whisk in milk and beaten egg.

Add all-purpose flour and buckwheat flour along with salt; whisk until smooth.

Whisk in melted butter.

Cover and let the batter rest for 30 minutes.

Heat a non-stick skillet over medium heat; brush with butter.

Pour a scant 1/3 cup of batter onto the skillet, swirling to coat the pan to form a blini.

Cook the blini until bubbles form on the surface and the bottom is browned, approximately 2 minutes.

Flip the blini carefully and cook until the other side is lightly browned, around 30 seconds.

Transfer the cooked blini to a baking sheet.

Repeat the process with the remaining batter, buttering the pan as needed and thinning the batter with milk if it becomes too thick.

Let the blini cool slightly.

In a bowl, whip the creme fraiche with an electric mixer until stiff peaks form.

Spread a thin layer of whipped creme fraiche on 8-9 blini and stack them, topping with a plain blini.

Trim off the rounded edges of the blini stack and cut into 1-inch squares.

Dip a layered side of each square into chopped chives.

Arrange the chive-coated squares on a platter.

Dollop caviar on top of each square and serve immediately.