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Preheat the oven to 350°F. Prepare an 8" square pan with a parchment sling or reusable silicone liner, then lightly grease the sides of the pan. Alternatively, line a 9" cake pan with a round piece of parchment or round reusable liner and lightly grease the sides.
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To make the filling: In a small bowl, combine the sugar and cocoa. Set aside.
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To make the crumbs: Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess. In a medium bowl, whisk together the flour, sugars, cocoa powder, and salt. Add the butter and vanilla; toss to combine. Using a bowl scraper, fork, or your fingertips, work in the butter until crumbs form and no dry spots remain. The crumbs should range in size from peas to walnuts. Stir in the chocolate chips. Set aside.
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To make the cake: In a large bowl or bowl of a stand mixer, combine the flour, sugar, cocoa, baking powder, salt, baking soda, and espresso powder. Add the butter and mix at low speed until the mixture looks sandy, about 1 to 2 minutes. (This is known as the reverse creaming or paste method.)
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Beat in the vanilla and eggs, then mix until no dry spots of flour remain. Stop the mixer and scrape the bowl and beater as needed; the mixture will be thick.
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Stir together the chocolate milk and vinegar. With the mixer running, slowly add the milk mixture. Mix until smooth, then stop the mixer to scrape the bowl and then mix again at medium-low speed for 1 minute; this final mix is key to ensuring the best texture so don't skip it.
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To assemble the coffee cake: Transfer half the batter (about 2 cups, 420g) into the prepared pan, spreading it all the way to the edges.
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Sprinkle the filling evenly over the batter, then sprinkle the chocolate chips over the filling.
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Pour the remaining batter over the filling and gently spread it to the edges (a small offset spatula is a helpful tool here). The top layer will be thin; it’s OK if a little bit of filling gets mixed in.
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Sprinkle the crumbs evenly over the batter and gently press to help them adhere.
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Bake the coffee cake for 45 to 50 minutes, until it's set in the middle and feels firm on top. When the cake is done, a toothpick or paring knife inserted into the center should come out mostly clean.
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Remove the coffee cake from the oven and allow it to cool in the pan for 20 minutes before cutting and serving. Serve warm or at room temperature.
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Storage information: Store leftover chocolate coffee cake, well wrapped, at room temperature for several days; freeze for longer storage.
