"Sludge" at the bottom of matcha/green tea latte

If you are making your drink with real matcha, that matcha is very finely ground green tea leaves. There are parts of the tea leaves that will always be insoluble in water, and therefore eventually fall to the bottom and create this sludge to which you are referring. The only solutions to that issue would be either 1) filtering the solution to remove the insoluble solids or 2) allowing the insoluble solids to settle to the bottom and the decanting the beverage above that sludge into another container. Straining is not enough when you are dealing with very fine particles, as the openings in strainers would just allow those particles to pass right through.

If, however, you are not using real matcha and it is completely water soluble (maybe a freeze-dried preparation from brewed tea) there will be no sludge. I would suspect that Starbucks is not brewing green tea with real matcha for every serving.

Use the milk frother but skip many of your steps (warm water unnecessary for example). First add the milk (oat milk works even better!) and turn on the frother, then, while the frother is on, add matcha powder gradually but make sure to finish a few seconds before the frother stops. If the powder has been exposed to moisture and is a bit clumpy you want to run it through a small sieve as you add it to the frother. I never add sweeteners so do not know when to add (perhaps try making one without sweetener to see if that helps), but I would do that after I have added the matcha to ensure the matcha is as well integrated as possible. I never have sludge except when I have clumpy matcha and no sieve. Most frothers have a cold whisk function you can use in case you did not finish all steps before the frother shuts off.

I make iced oat milk honey matcha lattes every day and don’t really have issues with sludge unless I leave my drink for a long period of time. My method is:

• microwave around 2 oz of water with 2-3 tsp honey for about 45 seconds (depends on microwave strength... can’t let it get to boiling heat or it will scorch the matcha and make it taste like an evil potion)
• while water is heating, I sift a teaspoon of matcha into a matcha chawan/bowl via an inexpensive stainless mesh matcha sifter
• pour the heated honey water (should be around 150-175 degrees, but under no circumstance should your milk or water be hotter than this for matcha) over the sifted matcha and whisk it rapidly with a matcha chasen for around a minute or two until light and frothy (there are tons of videos on whisking matcha out there)... proper aeration and separation of the little tea particles is your best defense against sludge
• shake up some oatly barista until it’s frothy and pour it over ice, and then pour the matcha over it; I also like to drink my lattes out of a mason jar with a Cuppow lid so I can just cover the top with my fingers and shake it up well before drinking. For hot lattes just pour the matcha over your appropriately heated milk.
• rinse off the matcha chasen and chawan with warm water, fill the chawan with warm water and whisk the chasen in it to clean off any residual stuff, dump everything out and lay it out to dry (benefit of using a chasen stand is that it can help ensure your chasen doesn’t get gross or moldy and will last a long time)

I really think your sludge issue is probably due to your whisking method, but if you are using low quality matcha (like culinary grade), I imagine those leaves might be a little heavier and more likely to create sludge at the bottom.

This is the chawan I have: Traditional Japanese Matcha Tea Cup Bowl | Multicolor Black Ceramic Ceremonial Chawan | Handcrafted, Lead Free, Large Macha Drinking Glass | Authentic Mug Shape https://smile.amazon.com/dp/B07Y1Z6SS3/ref=cm_sw_r_cp_api_fabc_Xy4PFbV5WPNGY I love that it’s shaped so that I can whisk matcha rapidly and not get matcha splashes anywhere

This is the chasen set I have: Matcha Tea Whisk Set - Bamboo Chasen and Black Whisk Holder Rest Set https://smile.amazon.com/dp/B07TKCQHHD/ref=cm_sw_r_cp_api_fabc_SD4PFbJH4D2W6 It whisks really well and the tines hold their curled shape for months and months so it does a beautiful job of aerating it