Making Lyophilized Cake Lookalike using household ingredients
You may want to consider whey. Looks like Karen Smith, Dairy Processing Technologist at the Wisconsin Center for Dairy Research, already did some of the work for you. The result depends on the specific type of whey (a high score of 4 or 5 on the caking test means the material cakes readily, forming a gummy crust): 




  Whey  – (Scored 2-5) –  whey exhibited a wide range of caking scores. How the whey is processed has a very large effect on the tendency of the resulting powder to cake as evident in this result. Clearly, two of the samples had large amounts of amorphous lactose and without the presence of significant amounts of protein the samples readily caked. 


Conclusion: choose a whey with less protein and more amorphous sugar (such as lactose). In general you are looking for a sticky substance. The amorphous sugar is sticky (it does not form a crystalline powder) and glues the material together. Bodybuilding whey may contain too much protein. Alternately milk powder (which contains plenty of lactose) may do the trick. I'm going to the market now to get supplies and run a test.

Update: I have run a home kitchen caking test on dried skim milk. The dry solids consist of 56% sugar, 40% protein, and ~1 g fat. The milk contains 11% water. 

The test was similar to that described in the linked document and consists of placing ~20 g of milk powder in a small (~3 inch diameter) aluminum baking dish and allowing it to float on hot water (~$\pu{50^\circ C}$) in a closed container for about an hour. 

The caking test is of course the opposite in a sense of lyophilization, for the solid is exposed to a humid atmosphere to encourage coalescence. The following image illustrates the result: 



This confirms the observation in the linked document that NFDM (lower left corner of the image array) "cakes upon storage due to presence of large amounts of amorphous lactose. The protein present limits the hardness of the cake of powder." 

This verifies that milk powder cakes. However I do not have a lyophilizer at home so unfortunately cannot verify that the dried milk will remain caked once freeze dried. I might yet put it in the oven to see how this affects the structure.